Carlyle St. Clair III can be found at his house. He was going to be the main course for Mortimer's banquet at the Ultra-Luxe, but when the White Glove Society attempted to kidnap Carlyle, he escaped. He hasn't left his house since the attempt.
Carlyle St. Clair's House is very well camouflaged against the mountain, and is difficult to find, especially if you are looking at every house in the vicinity. The shack is the same color as the mountain range and is flush up against the mountain. If you have trouble finding the shack where Carlyle St. Clair lives, just follow close to the edge of the east side of the mountain range, beginning just north of North Vegas Square.
"That obvious, huh? Yeah I'm not from around here. But where I'm from, they don't want me, so that's just as well. Really with my reputation, there aren't many places I can go that'll have me. My own family's blackballed me, and they've got a lot of clout."
pcps3xbox360 Once you get to the shack, he usually is not there. One solution has worked on PC, but you still may not have access to Carlyle's dialogue tree. Enter the following into the console: prid 127dd7, disable, enable. [verified]
Try knocking out Carlyle, drag his body to the dumpster, and see if the dumpster will let you put his body in. If this doesn't work, enter resetquest 0010d2c6 into the console just after the lines above and then go try knocking him out again after starting the quest over again. This means that you will have to go back to Mortimer and start the events over, and the characters that may have died previously will still be dead.
pcxbox360ps3 Choosing to pistol whip Carlyle St. Clair can sometimes kill him, causing you to immediately fail Beyond the Beef. [verified]