The best strategy for winning at Risk is to only engage in battles you are sure to win with minimal casualties - ironic, eh? There are a number of ways to do this; however, this is purely a heuristic, and it is not the only way to win. Risk is part formula, part thought, and you must adapt to the circumstances. This ability can be learned only through practice.
1. Initial placement. Do not put yourself in a position where you can be attacked from many directions. This means avoiding Asia, Europe (easily the worst continent on the board due to its relatively low worth compared to the number of places it can be attacked from), and perhaps Africa. Try to take North America, because it has few places to be attacked from, and gives a fairly large amount of men per turn. However, do not put up too much of a fight for North America in the beginning of the game, because once again, you want to get in as few battles as possible. In fact, you do not want to put yourself in any position in the beginning of the game in which you are fighting for control of a continent.
2. Fall back. If you do not want the enemy to take a territory, take a single territory of theirs in said country and pull your men back at the end of the turn. This will ensure that they will not get the bonus men that the country grants, and will also ensure that you do not get into a fight where you will lose a lot of men.
3. Buffer zones. Tempting as it may seem to place large amounts of men at the edge of your territories, this is a bad idea -unless you are desperate for the extra men a given continent will give you. You will most likely end up in a fight where you lose large amounts of men for no reason. Rather, after taking a continent, take a few territories in the adjacent continent around the entry point to your continent. Then pull back, leaving only one troop in each of those territories outside of your continent. This way, if an enemy attacks you, he will have to wade through a buffer zone, losing troops along the way until he stops or is weakened to a manageable size.
4. Let other people fight your battles. In the beginning of the game, and through much of the middle of it, there will be many fights as people try to gain continents. For the most part, you want to avoid this. Do not get stuck in the middle of a fight for a piece of territory, and gain alliances with the people near you - and try to make sure your allies do the fighting for you.
That's all for now. One last piece of advice: study the other players. Try to predict what they will do next. These strategies can often go down the drain if you are facing a suicidal player who will sacrifice all his men if he can get to you. Also, if you do not want the other players to gang up on you, try not to gain too much power if you don't think you can take them all on. Being the first to take a continent early in the game is a surefire way to end up at the top of everyone's hit list. Similarly, if you seriously hurt a player, make sure you can finish him off. If you hurt them, they will most likely carry a grudge against you.