Naughty Bear

I found the easiest way to do this was using a trap and scaring them then they become injured in which case come up behind them and it will give you the option to do an Ultra Kill (obviously make sure you don't have a weapon in your hand)

The easiest level to get this on is 1-2 Killer: Party Masacre challenge. This is only because the bears won't fight back as much, not because it is the first level.
Very obviously you do not want to kill the bears with a weapon so the goal is Ultra Kills or fisticuffs.
The trick to this is to leave things undestroyed. Whenever a bear runs to an item say either to escape or call the cops when you run up to them you will be given the option to give an Ultra Kill by pressing RT. Little things you can do too like sabotage machines and other objects to attract them to them to repair them, in which case you have the same Ultra Kill Option.
One way of getting around this though is through using a weapon, but when given the option to kill with the RT button discard the weaopn by pressing B and then finish them with the Ultra Kill, execution style.
This will take a bit of patience but can be very easy, just don't accidentaly over hit with your weapon and kill them if you are going to use the discard technique.

Very straight forward. I found this easiest if I didn't worry about points, trophies or combining with other achievements. Load up 1-1 Party Massacre, drop your weapon at the beginning and never pick another one up. To be safe, don't even punch another bear as I believe that counts as a "weapon". Then just use all your standard tricks of the trade - traps, sabotaged items and insanity kills. I found on this level they went insane very easily just by seeing all the dead bears.