I've always been told that a 1/4-1/3 of a couple of vinegar run through the entire coffee machine will clean it quite well and remove most stains. It's easy and worth a try. Remember to use plain old white vinegar.
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I've tried ice cubes and salt, as well as vinegar.

The best way I've found is dishwasher detergent.
Add a usual portion of whatever you use--1-2T or a tablet...and place in the pot. Fill with very hot water.
Let sit for an hour.
Rinse.

Repeat whenever you see stains.
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The way I was taught when I was working in a restaurant in college run by a Culinary Institute-trained chef is to put crushed ice, lemon juice, and kosher/rock salt into the pots and let them soak briefly (while I did another part of bar closing chores). After about 5 minutes, start swirling, swirling, swirling (the pots, not yourself, although if you enjoy being dizzy feel free) and then dump and rinse the pot. They should be clean and even disinfected using this method. Our pots were used for years, in constant use, and were never stained.
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A good old fashioned scrubby sponge and elbow grease works good for me.
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There are a variety of ways to get coffee stains out of the inside of your coffee pot.My favorite? #1 I take simple white/apple vinegar and mix it with warm water - and then use a sponge to clean out the coffee pot. This same mix works great if you're wanting to get the stains off your coffee pot maker as well. 
Other variations I've tried that work include:
#2 Substitute bleach for vinegar. As you  can imagine, the bleach cleans the coffee pot like no tomorrow...but then I spend a great deal of time washing the pot out a number of times to make sure I've cleared out the bleach. Thus, I opt for solution #1 unless it doesn't clear all the stains.
#3 I have used lemon juice and a sponge. If that doesn't work, I bring in the assistance of an SOS sponge pad. That's guaranteed to do the trick. Lemon juice is usually very effective. It's great if you need to get rid of bad fake tan spray lines...it's even better on coffee pots. :)
#4 Last but not least - if you've tried all the above and nothing is working...its time to decide - either the coffee stains add character to your pot and give your coffee extra history and flavoring...or it's time to give that pot to Goodwill and get another. I
Source(s):
personal experience

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Put ice chips, salt, lemon juice and the lemon wedges into the coffee pot.  Shake and swirl it around for a few minutes then rinse with warm water and wipe out with a paper towel
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