Basically, the random build challenge is exactly that - level a character from 1 to 60 with a completely randomized build. Open up the skill calculator on this site, and hit the chat gem at the bottom. The random build that pops up is the build for your character.
To make sure that you don't get stuck with a completely unworkable build, there's a few additional steps. First, if you don't have a basic attack for your class (resource generators for warrior, monk, or demon hunter, or signature spells for the wizard), note the build you got randomly, and roll up another random build. The first skill on the list that is a basic attack will replace the first skill you got on your actual build. Similarly, if you end up with any completely useless passives (such as something that boosts a skill that's not in your build), note your build and reroll, and replace the useless skill with the first one in the new build.
There's a few rules while playing, as well. First, no twinking - the only items you can use are those you find yourself. However, you are free to use gold from other characters, as well as crafting materials and potions, but you can't use anything off the AH.
When leveling up to 30, if you've got more unlocked skill slots than unlocked skills for your build, you can use any skill you do know to fill in the empty slot. Once you learn a skill from your build, you have to go back to town and change it as soon as possible. Similarly, if you haven't unlocked enough passives to fill your slots, you can pick any passive to use in its place until you learn the one from your build. If you don't have any extra slots but haven't unlocked a basic attack from your build, you can replace any one skill with any basic attack; similarly, if you have a passive that's currently useless because you don't have whatever it boosts, you can pick a different passive.
Runestones will be a bit of a challenge to figure out since we don't know exactly how they work yet, but while you're leveling, you have to either use the runestone from your random build or no runestone, and you have to equip a proper runestone as soon as you're able; if it's difficult to get a runestone for your ability, you're allowed to get a level 1 one off another character or the AH.
Once you hit level 30 and unlock everything from your build, you're locked in. Make sure your skills match the randomized build, and see how far you can get.
These characters should be treated as hardcore; once you die, you're done with that challenge. If you don't want to actually lose the character when you die however, feel free to play normal instead.
Let me know what you guys think!