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Legend has it that somewhere in Skyrim, is a hidden stash of catnip skooma. It's apparently protected by a herd of giant Skeever, and the brave Khajiit warrior must search skyrim singing his mating call at the top of his lungs from midnight to 3 am in order to pay bride-price to his prospective father-in-law back in Elsweyr. it is the Quest for the Holy Skoomanip...

You wish to play a Khajiit, no?

This one recommends you take a look: http://www.nexusmods.com/skyrim/mods/14513/?.

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Well, at least if you play on PC. I used it for my first Skyrim character(since I only bought the game about two months ago), and I don't think I could play a Khajiit without it. It has a few extra options if you have all the DLC, the Unofficial Patches and also for a few other mods(Live Another Life, The Paarthurnax Dilemma, Follower Commentary Overhaul, along with many others.

But my Khajjit is...well, I pretty much wrote an entire backstory for him before I even got the game. Ra'Maki is the child of a Khajiit Necromancer, whose story I played through in Oblivion(although more details than that are spoilers, but I was using two mods that deal with making a certain enemy in the game more powerful, and the other allowing you to pick a side after the questline relating to the mentioned character), and he(Ra'Maki), along with his mother, was kept alive through necromantic means, and were awoken a few months before Skyrim starts. After they make their trip into Skyrim, he and his mother are accosted by the Thalmor, who viewed their journey as highly suspicious. But rather than really wanting an explanation, one of them ends up killing his mother, which forces him into a rage. This leaves all of the thalmor dead, and he fights his sadness in order to give her a proper burial...which unfortunately means that while she did come back home to Skyrim, she would never live there again.

After this he makes the effort to find someplace to stay, only to find himself accosted again...this time by a group of Imperials. Throwing him into the back of a cart....eh, the rest is obvious.

This link has some info that might be of interest:

http://www.uesp.net/wiki/Lore:Khajiit

and this:

http://www.uesp.net/wiki/Lore:Khajiit_Names

Are you playing on PC? If you are, you might like to get the Khajiit follower INIGO. I highly recommend it, and he has a built in back-story for your character.

My previous Khajiit character was in Skyrim to find her brother (which was J'Zargo).

My last Khajiit was a mercenary, trying to make enough money to settle down, so I will suggest thinking what type of person is your Khajiit, what is their past, what do they wish to do, likes, dislikes? Think of them as a person instead just the stereotype of a race.

Try going along with the "not allowed into walled cities" rule, and see how much of the game you can do. I suspect you'll find there's quite a lot still there.

Khajiit heard that somebody wants to be a Khajiit. That made this Khajiit happy. But Khajiit warns you about nords, they are not nice to Khajiits.

Khajiit is done talking.

I usually have my last character be the parent/guardian/sibling etc to lay out a story on how the current character came to be. It's a good place to start.

Conjured bow hunter is pretty fun, a small backstory about visiting that mage outside of the college and him teaching taking you under his wing long enough to learn the bow spell. You rely on hunting for gold, do things a hunter would do. There are a few quests you have the option of doing. Think like a Khajiit when doing stuff, 'Oh, khajiit sees a horse with nice horns. Perhaps it has some nice skin to sell too', it'll add more of a connection to your PC. If you have access to mods, there are a LOT for a hunter playthrough.

Eventually you adopt 2 children and build a house for them. Bonus points if your next character is based of one of the children and have the story continue.