So I've started a new undead/vampire hunter build focusing on archery but I'm not sure what is stronger once improved with smithing perks and giving the crossbow a flame enchantment with the enchanting perk?

Personally, I do not like the crossbows, and go with improving Auriel's Bow, then I go to an ebony bow, then a daedric one, and finally a dragonbone bow.

I am also a bow afficianado.

Auriel's bow is a good one for taking down undead but it's all a matter of preference. Some stick with a lower quality bow to add challenge to the game.

If your'e like me and wish to continually improve your equipment, changing it out as you have access to better stuff then just take whichever has the best stats.

Personally I also like Ashley Clark ended up with a Dragon bone bow to which I added the chaos enchant to buff it up. It all depends on how you like to play the game.

Chaos enchant?

You can learn the enchantment on Solstheim by disenchanting a weapon you find. Chaos enchant works on a percentage chance per strike and can add fire/frost/electrical damage in any combination to a hit and it is more powerful the higher your enchanting skill. I took down a giant with five arrows using the chaos enchant on my bow with dragon bone arrows and the mentioned bow.

Edit: And I wasn't sneaking when I did it. I let Mjoll engage him and peppered him with arrows.

http://www.uesp.net/wiki/Dragonborn:Champion%27s_Cudgel#Champion.27s_Cudgel.

I always considered single enchantments better than chaos. If the math is right, shouldn't the damage from a single enchantment eventually overcome chaos?

You got me there DK.

Chaos should still be better for weaker targets though, at least until it reaches its threshold to the other enchantments.

At 37 per as it was with my just deleted character it adds up to 111 per shot which isn't too shabby.