My build is a bit like yours. You may want to check out my thread: LINK
I too wanted black, modern looking, straight edges, nice doors. My back is completely open.
What I did was measure everything, bought one really nice sheet of plywood from Lowe's, and had them cut it into "a bunch" of pieces.
If you look at the second to last pic on this page, you'll see what it ended up looking like:
LINK
I used Black Oil Based Rust-Oleum paint. Semi-Gloss is shiny enough. Could have used gloss. Very happy with how the paint has held up.
I scuffed up the doors before I painted them, because they had a clear finish.
In my case, to buy the doors, I found a "scratch and dent" vanity at Home Depot with perfect doors, marked down to about nothing, and bought it just for the doors. I disassembled it in the parking lot and took all the pieces home. I know you can buy doors online as well. Those doors are oak. Very solid, came with hinges attached. Made it easy.
As you can see from the thread, I used the scrap pieces of the plywood to build up an outer rim to give depth. Still straight lines, I think it looks nice.