Best method for rear venting a microwave on an interior wall?
From the clarification in the comments, you're lucky!

First, double check that your floor joists are lined up well enough that you can get a 10" duct through the floor where you need it. If a floor joist (or a beam) is massively in the way, you'll need some custom ducting or cancel the job entirely.

I'd cut a generous square at the bottom of the wall so you can cut out the plate in the stud bay. I'd cut slightly generously behind the range hood. (Give a few inches below the hood to allow a connection.) 

Make sure you don't have fire blocking/ diagonal bracing/ wiring/ plumbing in the way.

Assemble the hood with a short-way 90 degree elbow. Hang the hood, then push a length of assembled duct up to that elbow from the basement. 

If you're lucky again, your floor joists in the basement will let you run the ducting up in them. If the run can be straight out, you'll just need another short-way 90. Otherwise, if you have to zig-zag across the basement, you'll need a couple of long-way 45 degree elbows.

Typical ducting rules apply: use metal tape, screw where necessary, strap to support, and cut slices out of the corners of the duct to nest sections (don't waste your time crimping).

Having just done a very similar job, I can assure you that it's a pain. However, the result will be much better than recirculating.