How do you define sin nature? I believe in the concept of original sin inherited from Adam, but recently I was told that sin is actually in our blood. I have believed that sin is in our nature, but I'd never thought of it as actually being in our blood.

The best I know how to describe it, we are sinners in three aspects: sinners by representation, sinners by nature, and sinners by practice. Only the last of the three deals with sin as we commonly think of it. The first two are part of the curse brought on us by the fall of Adam in the Garden of Eden.

In Eden, Adam was the great champion of the human race. In Hebrews 7:9-10 [9] And as I may so say, Levi also, who receiveth tithes, payed tithes in Abraham. [10] For he was yet in the loins of his father, when Melchisedec met him.
See All..., Levi is said to have been in the loins of Abraham and paid tithes to Melchisidek even though he had not yet been born. In a similar way, I was in Adam when he disobeyed God and partook of the forbidden fruit. What he lost by that sin, I lost. He lost paradise on earth. I lost it. He lost eternal life and so did I. That is why Paul states that "in Adam all die" (1 Corinthians 15:22For as in Adam all die, even so in Christ shall all be made alive.
See All...). Romans 5:12Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:
See All... states further, "Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned." "For if through the offence of one many be dead" (Romans 5:15But not as the offence, so also is the free gift. For if through the offence of one many be dead, much more the grace of God, and the gift by grace, which is by one man, Jesus Christ, hath abounded unto many.
See All...) refers to the effects of Adam's sin on all. In 1Samuel 17, Goliath is the champion for the Philistines. He fought their battle for them. Therefore, "when the Philistines saw their champion was dead, they fled" (1 Samuel 17:51Therefore David ran, and stood upon the Philistine, and took his sword, and drew it out of the sheath thereof, and slew him, and cut off his head therewith. And when the Philistines saw their champion was dead, they fled.
See All...). When he lost, they lost. In like manner, when Adam sinned, we all suffered the consequences.

Something else happened when Adam sinned. His very nature was corrupted and that corrupted nature is passed on to all of his descendants. Adam was created in the image and likeness of God (Genesis 1:26-27 [26] And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth. [27] So God created man in his own image, in the image of God created he him; male and female created he them.
See All...). However, after the fall, Adam "begat a son in his own likeness, after his image; and called his name Seth" (Genesis 5:3And Adam lived an hundred and thirty years, and begat a son in his own likeness, and after his image; and called his name Seth:
See All...). Although we retain a shadow of the image of God in ourselves, our primary nature is the fallen nature of Adam. In the New Testament, the word "flesh" is often used to refer to this fallen nature (although flesh also refers to the material part of man and the meaty portion of the body according to context). Paul is referring to this sinful nature when he states, "For I know that in me (that is, in my flesh,) dwelleth no good thing" (Romans 7:18For I know that in me (that is, in my flesh,) dwelleth no good thing: for to will is present with me; but how to perform that which is good I find not.
See All...). Jesus came "in the likeness of sinful flesh" (Romans 8:3For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh:
See All...). The wording here is important. Naturally, the flesh referred to here is sinful. But in the case of Jesus, it was not sinful though it was visibly no different than all other flesh.

This refers to our practice of committing actual sins. There is no need to dwell here. However, I want to make an important point. It is this practice of sin that actually commits us to judgment in hell. The fall of Adam as our representative head condemned us to live in a fallen world. The corruption of Adam's nature by sin caused all of his descendants to be born with a corrupted nature that tends to sin and fails in many ways. But it is only our actual practice of sinning that condemns us to an eternal hell. This explains why infants who die do not go to hell.

You mentioned the teaching of sin being in the blood. I would disagree that sin or even the sin nature per se is in the blood. In fact, the blood of infants is called in Jeremiah 19:4Because they have forsaken me, and have estranged this place, and have burned incense in it unto other gods, whom neither they nor their fathers have known, nor the kings of Judah, and have filled this place with the blood of innocents;
See All... the "blood of innocents." If blood carries the guilt of sin from birth, then the blood of infants would not be innocent blood. It is true, though, that blood carries the guilt of those who shed innocent blood. "Thou art become guilty in thy blood that thou hast shed" (Ezekiel 22:4Thou art become guilty in thy blood that thou hast shed; and hast defiled thyself in thine idols which thou hast made; and thou hast caused thy days to draw near, and art come even unto thy years: therefore have I made thee a reproach unto the heathen, and a mocking to all countries.
See All...; see also Genesis 9:5-6 [5] And surely your blood of your lives will I require; at the hand of every beast will I require it, and at the hand of man; at the hand of every man's brother will I require the life of man. [6] Whoso sheddeth man's blood, by man shall his blood be shed: for in the image of God made he man.
See All...; Ezekiel 22:1-4 [1] Moreover the word of the LORD came unto me, saying, [2] Now, thou son of man, wilt thou judge, wilt thou judge the bloody city? yea, thou shalt shew her all her abominations. [3] Then say thou, Thus saith the Lord GOD, The city sheddeth blood in the midst of it, that her time may come, and maketh idols against herself to defile herself. [4] Thou art become guilty in thy blood that thou hast shed; and hast defiled thyself in thine idols which thou hast made; and thou hast caused thy days to draw near, and art come even unto thy years: therefore have I made thee a reproach unto the heathen, and a mocking to all countries.
See All..., 13).

On the other hand, the scriptures do seem to teach that blood is a carrier of corruption. Though Christ in His resurrected body had flesh and bones (Luke 24:39Behold my hands and my feet, that it is I myself: handle me, and see; for a spirit hath not flesh and bones, as ye see me have.
See All...), "flesh and blood cannot inherit the kingdom of God" (1 Corinthians 15:50Now this I say, brethren, that flesh and blood cannot inherit the kingdom of God; neither doth corruption inherit incorruption.
See All...). By comparing these two verses, we can see that the offending item is the blood. From a practical point, an embalmer quickly removes the blood from a dead body because it corrupts so quickly. In the Bible (Exodus 7:17-18 [17] Thus saith the LORD, In this thou shalt know that I am the LORD: behold, I will smite with the rod that is in mine hand upon the waters which are in the river, and they shall be turned to blood. [18] And the fish that is in the river shall die, and the river shall stink; and the Egyptians shall lothe to drink of the water of the river.
See All...; Revelation 11:6These have power to shut heaven, that it rain not in the days of their prophecy: and have power over waters to turn them to blood, and to smite the earth with all plagues, as often as they will.
See All...), the changing of water into blood was a corrupting plague. The blood is seen as something corrupt and unclean. The same could be said of how blood in treated in the law.

I know of no direct statement in scripture teaching that the sin nature is transmitted by the blood. However, many believe that it is. This is not to say that the sin nature is in the blood exclusively as something that dwells only in the blood. The AIDS virus affects the entire body, but can be transmitted to others through the blood. I think this is the concept most people have when they talk about the sin nature being transmitted through the blood.

This would explain some of the things that are said of the blood of Christ. His blood is called "the innocent blood" by Judas (Matthew 27:4Saying, I have sinned in that I have betrayed the innocent blood. And they said, What is that to us? see thou to that.
See All...). Although this phrase is found in other places in the Bible, it seems to be used in a special way here. The blood of Jesus was not just innocent; it was "the" innocent blood. Acts 20:28Take heed therefore unto yourselves, and to all the flock, over the which the Holy Ghost hath made you overseers, to feed the church of God, which he hath purchased with his own blood.
See All... tells us that God purchased the church "with his own blood." That is, it was the blood of God that paid for our sins. 1 Peter 1:18-19 [18] Forasmuch as ye know that ye were not redeemed with corruptible things, as silver and gold, from your vain conversation received by tradition from your fathers; [19] But with the precious blood of Christ, as of a lamb without blemish and without spot:
See All... tells us that we were not redeemed with corruptible things but with the "precious blood of Christ." Therefore, the blood of Christ was not corruptible. It was evidently divine blood.

This is how Jesus could be born in the "likeness of sinful flesh" (Romans 8:3For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh:
See All...), but without the sin nature. His flesh came from Adam, but His blood came from God. He experienced all the weaknesses of the physical body, but He did not have the sin nature. He came as the "last Adam" (1 Corinthians 15:45And so it is written, The first man Adam was made a living soul; the last Adam was made a quickening spirit.
See All...) and the "second man" (1 Corinthians 15:47The first man is of the earth, earthy; the second man is the Lord from heaven.
See All...). He came to win where Adam had lost.

I know there is a lot to digest here. I hope it has been a help to you. But through it all, we must remember that Jesus Christ came to give us the victory over sin and all its consequences. Because we are sinners by representation, we are subject to the presence of sin in the world. Because we are sinners by nature, we are subject to the power of sin in our lives. Because we are sinners by practice, we are subject to the penalty of sin. But Christ has redeemed us from all three: from the penalty of sin when we were saved; from the power of sin as we live for Him; and, from the presence of sin when we go to be with Him. Praise the Lord for His goodness!