My bathroom and kitchen are back to back, we want to extend our kitchen out by moving the bathroom back 4feet into a small bedroom. Our problem is the stack vent behind the wall we want to knock down to make the kitchen larger. How much of the stack vent will we have to tap into?

If I understand your problem you have a stack that will be out in the center of the room. Correct?
The stack will have to be cut up in the attic and offset into a plumbing wall. This stack will also have to off set under the floor.
However it may be possible to wye off the horizontal section to pick up your bathroom group.
Now tell me how the job will be vented.
Will a permit be pulled and the job inspected? Good luck, Tom