Bulgogi is surprisingly easy to make even without one of those in-table grills. Mmmmm, bulgogi.
Step 1: Slice beef
Get a nice big, SHARP knife, and cut the thinnest slices you can off of your hunk of meat, making sure to cut across the grain. (A good butcher can do this for you, but I don't always think that far ahead.)
Cutting the meat when it's well-chilled (or even frozen) helps to get nice even slices. Don't fret if you can't see through them; just do the best you can at cutting them thin.
The cut of meat you select isn't terribly important; just try to get a decently-sized cross section for your slices. I've also used bison roasts to good effect.