As close as they think can safely be achieved. I expect that will be in the range of hundreds of km, but that's purely speculation. It should be a fair bit closer than Pluto, however, because there they do not have to worry about flying too close to debris orbiting 2014-MU69. It's Hill Sphere is only about 400,000 km, and it seems likely that there wouldn't be any object of significance much closer. It seems the team hopes to have it's closest approach be less than 1000 km, from social media posts.
They have plenty of fuel to exact target it (within the uncertainty of it's position and New Horizons), and in fact, with the surplus of fuel they have, I expect it will basically target it spot on. Incidentally, the first burns have already started to aim New Horizons at 2014-MU69, as described in today's New Horizons mission news article. The biggest difficulty, at least initially, is likely targeting it precisely, as there is significant uncertainty in it's orbit. No doubt further studies, including from New Horizons, will help to refine its location.
EDIT It seems like the planned distance to target is about 1900 miles, per American Space. That is 4 times closer than Pluto's flyby.