Winning a free for all is largely a mater of diplomacy. If you act as though you want to take on everyone, everyone will be happy to make the game a 3v1 for a bit, since they're not going to be the 1. If you start being all campy, some play groups will ignore you in each person's own best interest, while others will gang up on you. When the game turns into two 1v1 games on opposite sides of the map, different groups have different standards for how you trade partners or switch to an unorganized free for all. Trying to make a switch in a way that not everyone agrees to can put you on the bad side of a 2v1 (while the last person charges up or starts spamming or just camps). Some groups follow a "gang up on the person who's in the lead" policy, while others tend to ignore such things.
So, in general, you have to learn how the people you play with play, and either exploit, adapt, or, in most cases, follow suit. And remember, as long as you win 25+% of the time, you're doing well. Some other, more specific notes:
Look for opportunities to make use of your opponents. If you grab someone, consider using him as a human shield for that projectile coming at you. Use your Kirby style AAA combo to paralyze someone long enough for someone else to smash him (just be careful you don't get hit instead!)
Speaking of grabs, learn which throws are safe. Some throws can damage other people in the area, and others come out fast enough to be safe. Bowser, for example, has a hard time in free for all partly because his throws (even side B) take too long to get going, so a third party can come in and hit both of you in the delay.
If person A is experiencing lag for some reason (shield broken, missed up-B as an attack, Jiggly's rest), and person B in a good position to punish, it's usually best to let person B punish before rushing into the area. Why make a risky attempt to attack someone when person B can do it for you?
Likewise, if someone is about to edgeguard/gimp someone else, you have an important decision to make: do you sit there and let the recovering person die, or do you interrupt and score a hit on the edgeguarder? Objectively speaking, the first option is usually better, but consider diplomacy. If the recovering person is about to be eliminated and you aren't sure you can beat the guarder given current damage/stock counts, you can often make a temporary ally by interrupting.
EDIT: I forgot something very important: Beware the desire for revenge! Chasing after someone because they did something to you will often lead you into a 2v1 or worse as people try to continue the normal game flow but periodically stop to deal with you. Trying to get revenge first on person A and then on person B in the same game is a sure way to fail.