All other answers are just totally incorrect…
Diamonds will randomly generate in the walls from y=0 to y=16. Lava generation is at y=11. Diamond generation does not get more frequent if you move further up, or down.
When you are beginning, the fastest way to acquire diamonds is to dig a hole down to y=13, then dig a 1x1x2 tunnel for as many blocks as you want, return to the start, move 5 blocks to the side, and repeat the process.
Once you've killed a wither and got a beacon, it's fastest to get eff V tools, set up a haste II beacon, and just completely destroy everything at y=12 and y=6.