To run a rift, first you must beat the game, and go to adventure mode, then beat 5 bounties, this gives you 5 rift pieces. Then you can open a rift (in any of the four towns) by going to the portal looking thing on the map. The rift will open and you'll be tasked to kill some number of enemies. (There is a progress bar, but not a specific number shown)
I believe you will have to kill 500 monsters on your first rift, just to get through it. If not, run a second rift and this should unlock. You do not have to kill the 500 in the same rift, it's over all rifts you play. After one rift I had unlocked this on my second rift the challenge screen told me I had already killed 1316 monsters, so it'll come very quickly.