Is it even possible? Stat wise I'm going for 1 health for every one magika.

Upgrading weapons is a must. Uprgading armour is sort of needed.
I like to play legendary with 100 health, 1-3 hit kill weapons, armour with no improvements and a follower with fully upgraded armour and powerul weapons.

Even with Destruction and Conjuration?

Impact perk for destruction but dont always stun-lock your enemies. Conjuration = Dremora Lords

No I'm asking even if I plan on using Destruction and Conjuration do I still need to improve my weapons?

It sure is. Just concentrate or sneak, illusion and conjuration. You should focus on perking the Illusion tree first so you can calm/fear anything you come across. For conjuration, try to get your hands on summon frost atronach (as a tank/obstacle) and summon bow (useful for increasing sneak skill through sneak attacks). You'll be set once you reach a critical sneak level, have silent casting/muffled steps and can cast ice atronachs repeatedly. Bring a follower and start casting frenzy and you'll have no problems at all. Equipment-wise, go for anything cloth or light that boosts magicka or reduces it cost - you should aim to not get hit or detected at all. Light equipment minimizes the speed penalty and should make dodging easier. Carry an extra bag of magicka potions just in case too.

When you can afford to, invest in alteration for detect life/undead and paralyze. Alternatively, use its Shout equivalent. If you do use shouts, Become Ethereal and Ice Form are excellent for escaping tight spots, while Throw Voice can be useful for clustering or luring out enemies for an illusion spell.

How about a character using one handed, destruction, light armor, restoration, conjuration, and alteration?

It's possible, but you'll pretty much rely on summons for your damage for most of the game, and maybe sneak attacks with a dagger(rinse repeat with pacify). Frenzy can help with multiple enemies too.