How do you make a diamond mine?
Dig down to the 11th level Y co-ordinate.
The 11th level being because that's the level with the highest Diamond Spawn rate.
Once there dig out an area to function as an underground base of operations.
The base should be minimum 4x4 (I suggest 6x6 or larger though), and have the following facilities.
1. Workbench
2. Furnace
3. Storage Chests (Optimally several)
In one of those chests you should have a few stacks on wood and torches in reserve, you'll need it.
Once your base is set up, start digging out tunnels from the base of operations (for ease of navigation I suggest in just two directions, may it be east and west or north and south).
Now with tunnels dug out, create small indents along with the wall, with three blocks distance in between:
* = Dug in
x = Blocks
***
x*x
x*x
x*x
***
x*x
x*x
x*x
(Base)
Now from those indents, continue to dig down them, making new tunnels branching off your current tunnels (and effectively making a Grid like system).
The reason you keep these tunnels 3 blocks seperate is because you can cover almost every vein of ore that way. Since most veins streach a minimum of 2 blocks across (meaning if one tunnel misses it the next one will catch it).
The only veins you might miss are those that are only a single block wide, in which case those are so tiny that going out of your way to get it will slow down your rate of resource gain.
And that's basically the trick in a nutshell.
Some notes though:
1. Main 2 tunnels place torches in the middle of the 3 block seperaters.
1.5 In the branching off tunnels, place torches every 5 blocks (basically everytime you mine out ahead).
This is to help the mine stay lit up, so the odds of Mob ambush is minimal, something you want when potentially working near lava.
2. When you feel you're done digging the tunnel, instead of running all the back dig four blocks to the left or right (in the direction you left the next indent) and then dig backwards.
This way even your return trip produces a haul.
3. When digging further away from your base place the torches to the right. When digging the return trip tunnel place torches to the left.
The reason for this for easy navigation, cause the grid system can get people lost easily otherwise.
If your torches are to the right it means you're heading further away from your base of operations, while to the left means you're heading closer to your base of operations.
Using the same trick in cave expeditions in general is also a good away to avoid getting lost when you get too adventurous and wander in really deep.
*Note if you're a more visual learner there's also a number of diamond mine video guides that can help you out.