I think it's 2 separate issues. People who say that "Black Lives Matter" should also be concerned about black-on-black crime are thinking about it in a pretty linear way. I'm sure they care about black-on-black crime.
The movement came about due to incidents where it was felt that police were specifically targeting blacks, and clearly was pent up anger over years of that kind of discrimination that many blacks - particularly those in the inner city - felt.
It's really the equivalent of saying that pro-life people should be trying to end war, or the death penalty. The pro-life movement is a specific response to abortion.