I have a Daedric Claymore and a black soul gem with a human soul in it and a few grand soul gems with grand souls in them. What should I enchant my sword with?

Soul Trap for sure although the 2nd part is up to you it could include any of the below

Drain Health
Damage Health
Fire Damage
Frost Damage
Lighting Damage

Probably Drain Health. I have yet to do the main quest, even though Daedra all pretty much have a weakness to shock I'll be fighting in dungeons against other stuff too. Should I enchant with max damage and whatnot?

I'd start with Soul Trap and then Absorb Health so that you get the benefit - as the enemy's health leaves him...you get it. I'd start there.

imo the best enchantment for a sword is elemental damage, absorb health, and soul trap. you can get more damage in with multiple damage types of small amounts, than one type of damage for a large amount.

Absorb health is my personal favorite.

Ok, question. How much of each should I dedicate the charge to. I can only put one gem into it right?

I enchanted a sword with a Grand Soul Gem with:

Soul Trap 2 sec
Fire Damage 5 pt for 1 sec
Frost Damage 5 pts for 1 sec
Absorb Health 5 pts for 1 sec

Charges up to 1600/1600 Uses: 57

I would have done Shock too but wasn't offered that one. I named it Say Hello To Heaven and it works great.

See more on this topic in the Spoilers forum at http://www.gamesas.com/index.php?/topic/1117614-weapon-enchantments/
I think there's some good advice there.

A couple considerations that are claymore specific.

1. Because of the long range of the claymore, the best fighting style is a quick and nimble character that can ably 'dance' right at the edge of the 'sweet spot' where the long range of the claymore can hit your enemy but your enemy cannot reach you. This takes some practice and is harder against some foes than others but nevertheless well worth developing if you like claymores (as I do). Ok, so where is this going? Well, absorb health is normally a wonderful enchantment, but I would not put it on a claymore. The reason is that I found myself constantly dancing in and out of melee range breaking the beam lock that must be maintained.

2. Because of the slower speed of the claymore, I would use 2 second enchantments instead of 1 second ones.

So. . . I would try soul trap x 2 sec + a rainbow of elemental damage effects all for 2 seconds each.

If you fight an enemy one on one depending on the enemy that you are fighting, Claymores are great to use. When you got 2 to 3 bandits or enemies attacking you the Claymore is still useful but you'll be taking damage unless you attack then move and attack again.

I don't really use claymores, but I would vary the enchantment depending on whether you use another weapon in conjunction with it or not.

seconded

So, based on the feedback, I have decided to go with a Daedric Longsword enchanted with: Absorb Health 5 pts for 1 sec, Fire Damage 5 pts for 1 sec, Frost Damage 5 pts for 1 sec, Shock Damage 5 pts for 1 sec, and Soultrap for 1 sec at the end.

Sounds good to me, good luck.
I personally just use goldbrand and always cast a silence/soultrap spell for 60 seconds after my first paralyze/weakness to fire spell for 10 seconds which has worked wonders for me.

Goldbrand huh? Hmm...I might have to go on a hunt for it. The "silcence/soultrap" spell you were talking about, did you make it where one spell does both?

Soul Trap
Fire Damage
Shock Damage
Turn Undead level 25
Light 10 ft for 1 second

Yup. I just use two hotkeys and I got them paralyzed and weakened(at least weakened for some creatures who resist paralysis) and I silence them and got their soul prepped. I use 60 and 10 but that requires being a Master in Illusion and Mysticism. You can just adjust the duration if you aren't a master in those skills. It can be kinda silly to silence non mages and paralyze those who can resist it, but it beats the heck out of casting 4 different spells(and using 4 hotkeys). I would do all in one except I don't always need their soul or to silence them in NPC cases. And I can paralyze multiple enemies quickly and not use up all of my magicka to do all 4 effects. All on touch by the way since its cheaper and more accurate.

On 360 btw so I only get 8 hotkeys.

I would do the simple Shock Damage 1 sec/Soul Trap 2 sec combo.

As you said, Daedra have a weakness to shock, and for almost anything else it's at least as good as any other element. Soul trap to recharge. I like to keep the enchantment fairly simply, so you aren't constantly recharging. Although if you have Azura's Star, making the enchantment a bit more complex might benefit you.

Ya that works pretty well. You could try a weakness to shock enchantment on it too but I'm not sure how the stacking and order works for that since I don't use custom weapons but I'm just throwing it out there.

Going with a long sword is a fine idea. They are good, flexible weapons that excell at sneak attacks and sword&shield fighting.

Your enchantment package sounds great. I'm assuming you will use a grand soul to enchant with. The maximum power per charge you can get is 85, and with 1600 total from the grand soul that gives you 19 charges. What I like to do is limit my charge power to as close to 75 as I can get it withough going over 75. Sure I get another couple charges, but I like the efficiency of recharging with this set up. If you go all the way up to 85, a petty soul (150) refills one charge and wastes the rest. With a power of 75 per charge that petty soul fills exactly two charges instead of one. Yes, very minor stuff I know, but that is what I do.

Ya longswords are great. You can always just use a sigil stone too and get more charges although you need multiple souls to refill it. It won't have any other enchantments but you can always just use the default soul trap spell at the end when their health is low.

I used a Black Soul Gem with a human soul in it to enchant my longsword. I think I get 57 uses out of my longsword, which I just got Azura's Star and put my empty Grand Soul Gems in my storage chest in my house in Bruma. So this way I know I have a spirit to use to charge up my weapon.

Yes. When using a blade that does not have soul trap on it (as is the case with so many of the in game blades, and certainly sigil enchanted blades), I use a companion spell to start melee combat:

Soul trap x 10 seconds + Weakness to magic 100 x 10 seconds. Helps any enchanted sword in the game be more usable.

Although soul trap is an excellent add on.....unless you have a plethora of gems lying around, you end up using up enchantment.

Absorb health with damage health is a nice one, as well as silence, depending on the character you are using. High Elves can benefit from this, if you are using one of course, due to low tolerances to magic. It will save you in the event you stumble onto a place with Necromancers, for example.

I might even save that soul gem for a lighter weapon or a bow with an area affect......

No I just use the Azura's star and whenever I capture a soul I just use the soul capture and keep fighting.