I used to self injure (primarily cutting, some burning), and it was related to my depression, not to suicide. However, when the depression worsened, then suicide became an issue. So even though the self harm itself was not directly related to suicide, there was the common denominator of depression, which is what had to be treated in order to minimize both the self injury and the suicidal ideation.
It has been over 8 years since I last self injured. Back then, I was in and out of hospitals almost constantly, and an occupational therapist kind of made it her mission to find an alterntive. As weird as it sounds, the alternative activity that worked was weaving. She made a very basic loom out of a piece of cardboard and used embroidery floss (like you make friendship bracelets out of), and when I felt like self injuring, I would weave until the urges passed. It was tactile, which I think is very important for most self injurers, and it required enough focus to occupy my thoughts, but not so much focus that I was unable to do it. I later moved on to Play Doh (which I considered a step down), which is nice because you can squeeze it, tear it, throw it, smash it, etc. Basically you can abuse the ball of Play Doh instead of abusing yourself. I used the Play Doh when I was working with teenagers and I had a couple of kids who self injured, and it was really effective. Knowing that you can beat the crap out of something in whatever way is most pleasing to you, and giving yourself permission to take those emotions out on something besides yourself, was very freeing for me and seemed to be for the kids I worked with.