I have a stealth archer that I'd like to make enchanted weapons for, specifically a dagger with soul trap and probably heavy shock damage just incase something gets too close. I have very little magic skill (between 5 and 15 in all areas), just wondering how I might go about doing that. I know I'd have to complete the mage guild recommendations which don't require magic to get access to the alters...but the enchanting part...

You can buy charged soul gems at most magical vendors and walaa. If you have the Wizard's Tower plugin you can enchant without getting into the Mage's Guild but you gotta pay a good chunk of cash to do it.

I think it isn't based on Magic skills when enchanting with soul gems. Just Spellmaking.
But I could be completely wrong on this one.

The only way magic affects enchanting is that you can only enchant items with effects that you've casted before. So if you don't have any shield spells, you can't enchant with shield. Unless you are using Sigil Stones but thats a whole different story.

Get like 10 paint brushes and hop the wall into the university.

Can you get into the Chironsaium and use the Altars still? Never tried it just curious.

Let's not get off topic here...

Can I enchant a weapon or armor without having the skill or mana to cast the spell?

No, sorry.

With a Sigil Stone yes otherwise nope.

You'll be able to learn a cheap shock spell even with minimum Destruction skill. The magnitude you can put on the weapon depends on the size of the soul stone you use to enchant it.

You much be able to cast Soul Trap to enchant a weapon with it at an enchanting altar. You need a Mysticism skill of 25. You can buy Minor Detect Life and cast it repeatedly - or make an even cheaper practice spell - until you get your skill up. This will take only a few minutes.

If you have rooted objections to raising your magic skill you can look for a sigil stone of soul trap, but you can place only one sigil stone on an object so you'd need separate weapons for the two enchantments you want.

Mara

Edit: you could get Umbra for the soul trap. Doesn't weigh anything until if/when you finish the quest.