There's 3 procedures that I've seen.
Getting debris out without changing oil to keep oil clean: With the oil still hot, drain the oil through a strainer into a metal pot. A finer strainer is better, but be careful not to let it overflow (i.e. don't walk away). Turn the fryer off as you're draining the oil or else it will smoke up. Depending on what type of floor you have, you might want to put towels or cardboard below the pot so the hot pot/oil don't melt the floor. You can use a metal brillo to scrub out any gunk left inside, and carefully ladle oil from the pot into the fryer to flush out anything left inside after the drain. Wait for the oil/pot to cool and then pour it back into the fryer after you close the drain.
Replacing the oil: Same procedure as above, except you don't have to strain the oil as you drain it because you'll be throwing it away. Before pouring in new oil you can use soapy water to clean out the inside, but make sure when you drain out the soapy water you're draining it into a different pot than you used for the oil. Also make sure to rinse out all soap with water, you don't want soap in the fryer when you put into new oil. As another poster said, make sure you strain out the soapy water when pouring it down the drain. Pour in new oil after you've dried out the inside of the frier, don't want to mix water with oil.
Deep clean: Drain out old oil as in first procedure, then fill the fryer with soapy water 2/3 of capacity after the fryer has cooled. Pouring water into a hot fryer can damage it and/or cause the metal within it to warp or crack. Turn frier on, let it stay hot for about 20-30 mins. Don't walk away, if the soapy water foams up and wants to overflow turn off the fryer. Turn off the fryer and let the water cool to where it's not scalding hot but also isn't cold. Scrub away at the inside. Drain, rinse, dry. Refill.
Safety tips:
1. Don't mix water with oil, especially hot oil. If you drop a bunch of water into hot oil it will flash to steam, exploding and sending hot oil everywhere. This is, of course, dangerous.
2. Don't ever try to move a pot of hot oil. Even if you think the oil is just warm, some of it still may be hot enough to burn you.
3. Make sure everyone around is aware that you're cleaning the fryer. When moving the oil make sure your path is unobstructed and that nobody is going to be in your way. Have someone hold open doors for you if necessary. Make sure the floor isn't slippery. Last thing you want to do is spill a whole pot of oil all over the floor. I've done it and it's hell on earth cleaning up large amounts of oil. You can't really mop it up. If you do spill it everywhere your best bet is to pour flour all over the oil and then scoop it up with a dustpan while you cry in shame and brood over your mistake.
4. If you do get burned by the oil, especially if it's a serious burn, don't rush to the sink and run the burn under cold water. Look up "degloving" for some nasty pics of why you don't do this. Essentially cold water on a hot oil burn will cause your skin to literally fall off. Let the burn cool, then flush it under lukewarm water. Then wrap the burn in gauze and treat as usual.
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