How to replace 12v halogen under cabinet lighting with LED lighting from scratch?
You can buy LED fixtures off the shelf, which are basically ready to go. I've seen these as puck-shaped lights that are basically designed for under-cabinet or shelf lights.



There are other pretty simple off-the-shelf solutions as well, such as using rope light.



If you are talking about using raw LEDs and literally doing everything yourself, it's pretty simple. You need to get some high output LEDs (pay attention to color temperature, beam angle, etc to make sure it is suitable). You can get LEDs in different packaging: clear/diffused, large/small, forward/side-emitting. Likely your best bet will be a large frosted one. How many you need will just depend on how much area you're covering, and how bright you want it.



You'll need a DC power supply. The exact values are all dependent on what you use, so it's very hard to give an answer here. There are online LED array calculators available, which can give you something like this: 



One of the major disadvantages of serial wiring (shown here) is that if one LED is disconnected or burns out, they all go out. Finding which one is broken can be difficult, and if you've ever done this with x-mas lights you'll understand what I mean. 



Another possible disadvantage of the DIY solution is that you are either going to have to come up with a nice way to package this all underneath, or you'll have a bunch of wires and bare LEDs hanging under your cabinets (maybe hidden by some moulding, but still). This may turn off future potential buyers. Off the top of my head, using some split-loom tubing may be a way to hide the wires and make it look at least half-decent.