Remember you can use axes as weapons as well, a wooden axe will save the day against an unarmed player.
Don't get put off by armor, only by weapons; if you're unarmored but have any weapon and you encounter an armored player with no weapon, consider him prey and chase him down at any cost, since anything short of full iron armor won't cut the edge of a clear weapon advantage.
Keep an eye out for players with pieces of golden or iron armor, they'll generally have obtained it from the cornucopia (the center), and with it they may have collected some diamonds, gold or iron ingots or cooked food.
Don't start fights too early; fisticuffs with another player right after getting out of the park may result in reduced competition, but it'll generally 1. waste time, 2. yield no loot and 3. leave you vulnerable. If someone's headed the same way as you, just run parallel to him and ignore him till you have a weapon to finish him off with.
Know when to stop looting. Even if there are still unopened or refilled chests around, remember you'll only get mediocre gear and, on a rather low percentage, iron bars from them. I'd recommend to stop looting once you have a stone sword if you didn't find any iron bars yet, and if you did, keep looting till you can make an iron sword. Then, it's time to hunt for the players who got random pieces of gear from the cornucopia.
And lastly, stay on the move. Camping or hiding may seem a good idea when it gets you in the top 3, but it's deceptive: while you hide in your hole, all the gear collected by all the other players will be gathered by the ones who kill them, and eventually it will distil down into one badass with full iron armor and a diamond sword, leaving you no chance whatsoever in the final deathmatch. Stop this by raiding the cornucopia periodically (as it gets refilled) and killing players yourself.