This is one of the hardest challenges of the game. You have 1:36 to get to the bottom while destroying 10 snowmen. The biggest problem here is if you don't get a flawless run, you won't make it to the bottom of the slope even if you don't try to destroy snowmen.
There is 20 snowmen on the course. At least you have some leeway on which one to destroy. I also noticed that near the end, the snowmen are easier to reach.
I would not recommend trying to get snowmen out of your way. You don't have the time to do so. Simply take the ones in your path. Also, I would recommend going into first-person mode (by pressing B). That way, your body won't hide the incoming snowmen.
This will definitely take a lot of your time. Persevere, and try to find the best snowmen to destroy while not losing much time.