I am looking for an assassin character who dual-wields daggers. Problem is that, at least early in game, the daggers are ridiculously weak. So, where can I obtain two decent daggers really early within the game?

do the mehrunes dagon and Dark brotherhood quests

You get a good starting dagger from the Blessings of Nature quest in Whiterun. I don't think that there are any level requirements for starting. I'm not sure about any others though. Best of luck- assassin is a fun build!

You could power level your smithing to make a better daggers.

This is the easiest thing I can think of.

@Nopenske You need to be level 20 to get Mehrunes Razor but doing the DB quests would work.

Power leveling is the easiest and quickest (off the top of my head) to do since finding iron ore is easy to do.

Just smith tons of iron daggers and you'll be on your way to getting a good dagger.

There are two elven daggers in a locked display case in the thalmor embassy.

At lvl1 a quick and easy dagger find is to clear Moss Mother Cavern for Valdr, just West of Lake Ilinalta.
You'll get Valdr's Lucky Dagger which, though only Steel, has a 25% chance of dealing a critical hit.

Walkthrough if needed:
http://www.uesp.net/wiki/Skyrim:Moss_Mother_Cavern_%28quest%29

That critical hit percentage is quite nice and you can improve it with the simple Steel smithing perk; no Arcane smithing needed.
Hope that helps

Edit: Although this has absolutely nothing at all to do with Dawnguard.

+1 I used this for quite some levels.

Like other people mentioned, power leveling is just your best bet. As an assassin, you will want to be doing sneak attacks anyways. If you want to power level sneak abilities, get the thief stone. Then get in with the dark brotherhood and get their gloves for 2x backstab damage. Get the perk that gives you 15x damage and you're doing 30x damage on a sneak attack. That will kill just about anything instantly. In the meantime, work on getting enchanted equipment. Upgrading my equipment with magical proprieties helped me level even faster. Time the shadow warrior right and you can sneak attack while in combat. I've read that even the lowly iron dagger can own in the right hands. I never bothered to try, I like my fancy toys.

That's cruel, that hagraven destroys low leveled people with its nuke spells.

If you've leveled pickpocketing and gotten the appropriate perks, you can

If you take the Sneak Perk "Assassin's Blade" and pump a couple of points into the One-handed Skill, even a Steel Dagger will cause monumental damage.

And since you're playing an Assassin, you should be heavily investing in Sneak anyway, as well as One-handed and Archery.

IMO of course.

How about just play the game without power leveling or getting the best weapons straight away.

You'll just have to work you way up with either the occasional smithing or level up. Its not quick unless you heavily train skills or power level.

You'd have to smith iron or steel in the beginning. Or maybe dwarven if you level up good enough.