I am basically looking for the best race for archery in Skyrim. I am looking for a fast nimble character that can deal quick high damage. Also, what would be the best weapon/gear for such character.
As already commented, you can max out or grind any skills with any race in Skyrim, so choice of race for archery won't really matter in the long run.
If you want to have an additional boost for archery at the early parts of the game, the Bosmer (Wood Elves) are strongly suited for archery. They have a +10 bonus to the archery skill and a +5 bonus to sneak (to help deal sneak attacks).
If roleplaying, the Bosmer are known as being "the best archers in all of Tamriel, the Bosmer snatch and fire arrows in one continuous motion; they are even rumored to have invented the bow". In-game description describes them as making "good scouts and thieves, and there are no finer archers in all of Tamriel".
I am looking for a fast nimble character
Well, you have three choices. High Elf, Orc or Nord.
Due to the way that character size interacts with movement speed, taller characters move faster than smaller characters.
8% faster High Elf (m/f) 4% faster Orc (m/f) - rage affects bow damage 3% faster Nord (m/f) - cold resist is helpful baseline Bosmer (f) 2% slower Bosmer (m) - oops.
Any race will allow you to become a Master-level archer, if you work at it enough. That said, I believe the Bosmer (wood elf) race is the only one to give you a +10 boost at the beginning of the game, meaning you'd have to level that skill that many fewer times to reach the maximum capability.
The Khajit and Redguard races give +5 archery to begin with, making those good secondary classes to use.
Cheap answer: Bosmer because of the high startingbonus. But that doesn't really matter. You can take what you want. Every race has the maximum of Level 100.
In my opinion, the Khajiit fits the most, because they get bonus in sneaking, archery, alchemy, pickpocketing and lockpicking.
They're feared as silent thieves and robbers in Skyrim. From the atmosphere, it would be the preffered race for an "archer". :)