So I made a Breton character with the intention of making her into a Destruction oriented Battle Mage, I noticed she's quite low on the Destruction stats though, can I still mold her into a kickass Destruction mage, or do I have to choose a different race?

totally doable, race is only a minor factor. play which ever race you feel the most comfortable, especially when you will be spending hours with him/her

absolute no Problem Mate. the skill will raise like mad as you flame trough Doungeons

The current issue with resistances aside you'll be fine as a battlemage. Breton or High Elf probably make the best initial ones though Dunmer are also good. My biggest piece of advice is even if you decide to power level smithing/enchantment. Which btw you wont really need to do to stay competitive with creatures in the world you're better off NOT wearing armor early in the game. Bound weapons are great though and until you can start making awesome enchanted glass or daedric weapons bound weapons are one of your best damage spells for a large part of the game.

Which is the only downside. A bound sword with the damage upgrade perk is so much better than most destruction spells it's a bit sad.