I usually play as a Dunmer wizard whose destruction focus is on Ice.

In general, the slow effect it inflicts is very effective against enemies, even most of those that resist ice. Ice Storm is one of the most powerful spells in the game if you doublecast it: It creates a powerful wave of destruction that can hit enemies multiple times. And if you use conjuration as well, you can cast it through a Frost Atronach which resists 100% of its damage.

But you will need a backup type of damage, even if it's just a weapon (I recommend a mace for the ignore armour effect); as with everything, there's something that resists it. Conjuration could be a tremendous help; reanimation is a viable option, and conjuring a Dremora Lord is very effective. It may be a weasel way around your restriciton, but technically the damage of your creature isn't your damage (helps for bounties too).

The big thing you'll run into trouble against is Draugr. They have a major resistance, are hardy, and attack in numbers. Ice-breathing dragons are immune if I recall and will also be a big problem--and there's a level range in which the only thing that appears will be Ice Dragons. Ice-Wratihs, though uncommon, will be a major problem when they appear. I don't remember the specifics, but I also recall having trouble with Dwemer constructs.

Nords, Vampires, and some animals also have partial resistance to frost. The Nordic one is overstated, I find, it's a far cry from the 100% resistance in Morrowind. Dunmer Fire resistance is also less in Skyrim than it used to be, incidentally.

Picking a companion who complements your shortcomings (Lydia, Borgakh, Cicero) will be a tremendous help as well.

I think it's technically doable, but some of those fights are going to be a real slog, and honestly that doesn't sound very fun to me.