It really varies - you should expect, on average, for the temperature to be usually not much lower than -5 F or thereabouts. It may get down to -20 F, but that would be pretty rare, especially for more than one consecutive day. One of the good things is that we've got lots of sunny weather, so even it it is -5 F, it doesn't usually feel like it. Lake Louise also gets lots of inversions in cold weather, so it will sometimes be -5 F on the bottom and like 20 F on top. It can get windblown on the backside though.
For my body, I dress like most people usually. I have a thin merino wool top I wear and lightweight merino bottoms. If it gets lower than -5 f, I have a heavyweight pair of merino bottoms from icebreaker that I bust out. On top of that, I will wear a fleece hoodie or my Patagonia Nano Puff on top, and then I've got a Marmot insulated jacket and insulated snow pants. When it's cold, I'll wear a thin merino liner under my helmet, and I will wear thin merino wool glove liners underneath my insulated mittens (I prefer mittens when it's cold). I wear midweight wool ski socks. I always keep a fleece buff with me to pull up over parts of my face. This is especially useful at Sunshine Village, where you're often above tree line. So, not very different from most places.
The bigger challenge I have is not staying warm, but staying dry. Even when it is that cold, I will sweat, but the problem is when it's that cold, you feel wet clothes way more than usual. So, I actually will often have some extra layers to change out at lunch time. I'll usually bring another wool top. I find the thing I change out almost all the time are my glove liners and mittens. If there's even a little bit of moisture, I won't be able to keep my hands as warm as I like them, so I always change it out. Keeping my hands warm on a cold day is, for me, the biggest challenge. I have never used the chemical handwarmers, but I know a lot of people that do.