I struggle with this and so does my best friend.
Hi, becca4peace.
Since you asked what the Bible has to say about "cutting/self harm", I'll begin my response with this:
Mark chapter 5
[1] And they came over unto the other side of the sea, into the country of the Gadarenes.
[2] And when he was come out of the ship, immediately there met him out of the tombs a man with an unclean spirit,
[3] Who had his dwelling among the tombs; and no man could bind him, no, not with chains:
[4] Because that he had been often bound with fetters and chains, and the chains had been plucked asunder by him, and the fetters broken in pieces: neither could any man tame him.
[5] And always, night and day, he was in the mountains, and in the tombs, crying, and cutting himself with stones.
[6] But when he saw Jesus afar off, he ran and worshipped him,
[7] And cried with a loud voice, and said, What have I to do with thee, Jesus, thou Son of the most high God? I adjure thee by God, that thou torment me not.
[8] For he said unto him, Come out of the man, thou unclean spirit.
[9] And he asked him, What is thy name? And he answered, saying, My name is Legion: for we are many.
[10] And he besought him much that he would not send them away out of the country.
[11] Now there was there nigh unto the mountains a great herd of swine feeding.
[12] And all the devils besought him, saying, Send us into the swine, that we may enter into them.
[13] And forthwith Jesus gave them leave. And the unclean spirits went out, and entered into the swine: and the herd ran violently down a steep place into the sea, (they were about two thousand): and were choked in the sea.
[14] And they that fed the swine fled, and told it in the city, and in the country. And they went out to see what it was that was done.
[15] And they come to Jesus, and see him that was possessed with the devil, and had the legion, sitting, and clothed, and in his right mind: and they were afraid.
[16] And they that saw it told them how it befell to him that was possessed with the devil, and also concerning the swine.
[17] And they began to pray him to depart out of their coasts.
[18] And when he was come into the ship, he that had been possessed with the devil prayed him that he might be with him.
[19] Howbeit Jesus suffered him not, but saith unto him, Go home to thy friends, and tell them how great things the Lord hath done for thee, and hath had compassion on thee.
In this man's particular case, there definitely was demonic influence which led to his "cutting himself".
That being said, this particular topic is very close to my own heart. You see, I have a twenty year old niece who had been (she's recently stopped) cutting herself for quite a bit of time and quite severely at that. She's actually wound up in the hospital in the past with self-inflicted cuts all over her face, arms, legs and body. In her particular case, seeing how I've known her personally for her entire life and seeing how I therefore know quite a bit about her own struggles via my own observations, her own testimony and the testimonies of her parents (my sister and brother-in-law), there definitely seems to be a strong connection between rejection, whether genuine or wrongly perceived, and her cutting. IOW, seeing how she's been rejected (again, whether genuinely or wrongly perceived) by so many different people at so many different stages of her life, she's basically turned to self-harm as some sort of punishment.
Anyhow, the cure to the same, in my estimation, is finding acceptance and forgiveness in/through Christ Jesus...and I'd be happy to discuss this with you (and/or others) if you're but interested in the same.
