Any knife related hunting Challenges are tricky, but doable...
The Cougar and the Bear's tactics are nearly identical, with an obvious difference in speed. They close in for an attack, swoop out, will pad around watching you, then go for a second sweeping run. Both animals WILL toss you around like a wet rag if they make contact, and can usually hit you soon enough after the first strike to kill you.
HOWEVER! If you stand there and let them come, then roll to the side at the last second, you won't be knocked down, even if the attack gets you. You'll have a precious second to slash out at it's side before it gets out of range.
The Cougar is pretty easy once ya do this. I managed it without resorting to cheap tactics like shooting it in the paw :/ But the Bear... that's another story. I tried to do that one the honest way, but I was getting smacked around like a bowling pin, so I shot it a few times first.
If ya get to the Kill 3 Bears each with one shot Challenge, use the throwing knife. Instant kill if it hits the head.