Make sure you ended on good terms.
In order to remain friends after a break up, make sure the breakup itself was neither nasty or demeaning. Rarely are relationship endings mutual, so it's important to break up with someone, or take the break up, in a mature, reasonable fashion.

You should also consider the reason why you broke up before deciding to be friends. If you broke up with a boyfriend because he hit you, or if you ended things with your girlfriend because she was constantly accusing you of flirting with other girls, then perhaps it might be best to go your separate ways.

Give it time.
A breakup can be a painful experience, which is why it is important to give things time before jumping into a friendship. Giving it a couple weeks or more is a good idea to let things cool down, not only the pain of the breakup but also the physical desires that may still linger after a breakup. Keeping your distance for a little while will allow you to get used to being single instead of a couple again and help you to avoid making any mistakes. Once you have done that, you can honestly assess what kind of friendship you want and whether you even want one anymore; some people decide that it's too painful to remain friends.

Consider the reason for wanting to be friends.
If you want to remain friends with your ex, you should do so for the right reasons. Examples of the wrong reasons include hanging on with the hopes that the two of you will get back together or wanting to keep tabs on the person and who they date next. However, if you genuinely enjoy the person's company and care about them in a non romantic way, then staying friends is not a bad idea.

If there were children involved in your relationship, it's healthy for all involved if you remain friendly and civil with each other, even if you have no desire to be friends with your ex.

Make sure your understanding of the friendship is clear.
It is important to know the boundaries of your friendship, since it can be easy to feel like a couple again. Know where you stand in terms of calling and talking on the phone, hanging out, and topics of discussion. Often times, two people's definition of remaining friends is two completely different things. One person may think staying friends means saying hi when you bump into each other, while the other person may think it means calling to chat or meeting up for lunch every week.

Know that eventually, the friendship will change.
Even if you decide to stay friends, eventually that relationship will change as you both begin dating other people. Certain topics may become taboo in your relationship as well, so it's a good idea to enter into your new platonic friendship with realistic expectations about its outcome.