When it comes to stalking and evasion, cougars are geniuses. Bears and wolves are probably easier to hunt than cougars. The easiest way to hunt them is with traps or dogs, neither of which are all that ethical.

Anyways down to my advice. You'll want to be 'invisible' for this. Don't use any conditioners or smelly soaps when showering beforehand, rub pine needles/native vegitation all over your clothing til you reek of it, wear plenty of camo, no smoking. First you have to be lucky enough to find a cougar track. Move slowly and stay low/in cover, and try to stay downwind of where you think the cougar is. Try to get an experienced tracker or guide to go with you, as you'll want a partner anyways. Have a high caliber pistol around just in case (like .44 or .357). Cougars aren't particularly aggressive (unless wounded), but they are damn smart and there's a chance you won't see it til you're only a couple meters away. There's a chance that when you do find it, it will already know you're there and is in fact stalking you.

Don't feel bad if you don't get a clean kill shot, but you'll want to kill it as fast as possible. Cougars are made of %50 muscle, %20 adrenaline, %5 genius, %15 evil, and %10 holocaust. They are the serial killers of the woods, and aren't particularly safe to hunt when wounded.

Then you can eat it's heart and stuff.