How to Win at Risk
Try to make yourself as strong as possible. Don't let opponents attack you; let them kill each other instead. Strengthen your borders and make yourself impenetrable. I find I do best when I take Australia, and put almost all my troops in the Middle East. The Middle East borders Africa, Europe, and it's in Asia, so you can "police the world" in a sense. With your bonus from Australia, you can usually keep troops in the Middle East pretty easily. If no one can attack your powerful borders, they'll be attacking each other, thus weakening themselves. If they're at a standstill while you're getting stronger, you win. Plank the ledge in an FFA. North America is the biggest threat to this strategy, so try to keep pressure with Kamchatka as well. When you are strong enough go in for the kill.
Tips:
1. If a huge enemy army is approaching, don't attack an enemy territory next to it. Let the other guy take the beating for you.
2. Don't start out competing for continents. Even though Australia might be the best, don't put all your resources there with someone else. You'll just end up killing each other and getting wiped out by the other opponents.
3. Pay attention to the amount of borders your continent has. While Europe gives 5 bonus for 7 territories, the best ratio in the game, it can be attacked from 4 different places, and from three different continents. Remember, your goal is to avoid battles, not be in the middle of them. Australia only has one border, so it's relatively easy to defend.
4. Don't attack a territory unless you have more troops than it does. You will probably lose and instead of leaving yourself weak to one opponent, you leave yourself weak to multiple opponents.
5. Even if an enemy territory has more troops than you, it might not attack you in fear of leaving itself open. They might want to avoid big battles too.
TL;DR ----- Conserve troops by making borders strong and avoiding battles - Take Australia and police the world with the Middle East.