My kitchenAid stand mixer is leaking oil. How do I

Yes you can. Here's the link to an exploded diagram, and parts list.
You'll need to disassemble the mixer down to the bad part, order and replace the defective piece and reinstall it. I always recommend getting the exploded diagram first as it makes the process much easier, especially for the first time.

http://www.ereplacementparts.com/kitchenaid-kp2671-stand-mixer-parts-c-114958_114959_116129.html

Best regards, --W/D--
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It's not oil, it's grease. Here is a very detailed article on how to add grease to the mixer. After you have added some new grease and tightened the screws it will stop leaking. The temporary fix is to simply remove the chrome ring as is done in the picture on the attached link, clean out the oil and stick it back on till it fills up again. It is called a drip ring for the reason that it catches the oil component of the grease which drips from the gasket when the screws loosen:
http://www.artisanbreadbaking.com/discussions/kitchen-aid.htm

Leaking of oil in a stand mixer is not necessarily indicative of a large problem.

Kitchenaid stand mixers are overpacked with oil to last the lifetime of the motor. If the motor sits idle for some time (is not used), the oil may begin to drip from the gears and settle. Oil leakage may primarily be seen from around the beater shaft or planetary action.

A stand mixer can lose up to a 1/4 cup of oil before it needs to be serviced.

It is recommended to run the mixer on speed 10 for 2 minutes in order to redistribute oil back into the motor. In order to prevent future occurences, this is recommended to be done every 3 weeks if mixer is not being used.

take off the chrome ring around the rotator. remove the screws that were hiding under it. pull off the rotator and replace the "O" ring on the shaft.

It can be. This may require replaing the plate that the shaft comes through, since there is now enough play to allow the grease that is in the nead to leak through. You can find the parts online, but be ware, it involves tapping out pins and undoing many screws. Manuals are available online at Kitchenaid.com

There are screws under that metallic ring on the mixer head. Gently remove the metalic ring by pushing down on its edge with a screwdriver. Once the ring is removed the screws are exposed. Tighten each one and then again to make sure they are snug. If this doesn't work, you need a new grease gasket. Good luck.

The "wobbling" sounds like an issue of beater to bowl adjustment, which can come out of alignment naturally over time and with use of the machine.

If the flat beater is striking the bottom of your bowl, this may cause the locking mechanism to loosen and may cause the mixing head to "bounce" during use.

Your Use and Care Guide will cover the "beater to bowl adjustment," which you can do at home in a few minutes with a flathead screwdriver.

Also, leaking of oil in a stand mixer is not necessarily indicative of a large problem.

Kitchenaid stand mixers are overpacked with oil to last the lifetime of the motor. If the motor sits idle for some time (is not used), the oil may begin to drip from the gears and settle. Oil leakage may primarily be seen from around the beater shaft or planetary action.

A stand mixer can lose up to a 1/4 cup of oil before it needs to be serviced.

It is recommended to run the mixer on speed 10 for 2 minutes in order to redistribute oil back into the motor. In order to prevent future occurences, this is recommended to be done every 3 weeks if mixer is not being used.

Found this,may be helpful.
Why do I see oil leaking from the beater shaft of my stand mixer?
Some KitchenAid® stand mixers that are not used very often or have been tipped may experience slight oil leakage from the shaft or hub attachment. To avoid the oil leakage, KitchenAid suggests that you run the mixer weekly for short periods of time. Generally, the mixer may lose approximately X cup of oil before repair is necessary. The oil used is FDA approved and is not harmful to consume, should any get into the food that you are preparing.
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It depends on how mechanical you are. The old part will probably need to be cut off since it is pressed into the base of the rotating head. Parts are available from many sites like partstore.com and the like.

Leaking of oil in a stand mixer is not necessarily indicative of a large problem.

Kitchenaid stand mixers are overpacked with oil to last the lifetime of the motor. If the motor sits idle for some time (is not used), the oil may begin to drip from the gears and settle. Oil leakage may primarily be seen from around the beater shaft or planetary action.

A stand mixer can lose up to a 1/4 cup of oil before it needs to be serviced.

It is recommended to run the mixer on speed 10 for 2 minutes in order to redistribute oil back into the motor. In order to prevent future occurences, this is recommended to be done every 3 weeks if mixer is not being used.
Exploded parts views for all mixers are available at www.kithchenaid.com. Click "Customer Care" in the upper right hand corner of the main page, and then select "Locate Manuals and Guides." You will need to enter the unit's model number exactly as it appears on the bottom of the mixer. It will be listed on a white sticker.