I've been thinking about making a mage character, and do you guys think it would be possible to go through the game using just magicka? no swords or bows or any other weapons, just magicka.(oh and bound sword would be fine)

bound sword is still a sword

Yes but it increases conjuration.(therefore I wouldn't mind using it)

Depends on what difficulty level you play on, one novice difficulty it's going to be extremely easy to use only magic.
On harder levels it'll still be possible, summon a frost atronarch, stun enemies with impact etc.

So yes, it's possible.

Adept difficulty is ridiculously easy.
Sure, a Pure Mage could be okay!
But early on, say trying to use the Gold Claw right at the beginning when you're Level 2, it's gonna be tough.
Conjuration would do much more for you than Destruction.
Conjure critters, and conjure bound weapons. Train the Destruction!
Fun.
Good luck with that.
I don't know if it could be done without conjured weapons...

Sure you can. The easiest way is to reduce the costs of your Detruction spells to 0 and just Dual-Cast them.

It is absolutely possible. With no armor, and no weapons. On master level. The "paralyze" spell in Alteration is the most powerful spell in the game. It even effects every mob in the game. Some would call it overpowered. It might ruin your fun. Illusion school rage/fury/mayhem + invisibility/muffle others would say is another "so powerful it ruins your fun" combination. Most important...get Destruction up to 40, somehow, and take the Impact perk in the Destruction tree.

It is definitely possible! If you train high enough, the master level spells that you can use will be extremely helpful. Just remember that every magic school is good in its own way, use them in the right place at the right time and magic is extremely powerful. Also, make sure to keep on top of the spells, I.e If you are still using flames at level 30, harder enemies will be much more resistant. It might sound simple, but I switched to destruction magic at level 36 and it wasn't easy