DON'T remove the back. Fiddle repairs almost always involve removing the top. Now, since you're dealing with a broken heel-block, the danger of removing the back is lessened, but in most cases, the integrity of the heelblock-to-back joint is very important, and one does not typically mess with it unless it can't be helped.

If the block can be repaired in place, the button can be grafted onto the back and then the neck, when it fits back into the dove-tail, can be glued to the button.

If this fiddle is a decent one, and your description sounds like it might well be, I'd take it to a real violin shop for at least an estimate. And ask them about the techniques you'd need to use if you do it yourself. For that matter, ask them whether the back or top should be the default removal choice for repairs.

Bob