All it takes to get to level 1 is to beat one or two games of each type.
The quickest way to win a game of minesweeper is to play on easy. Your first move will never hit a mine, and should clear quite a few spaces. There are some minesweeper solvers available, but I didn't find one that works well enough to recommend. I found that the Normal level was the best way for me to accumulate points quickly. I could almost always win, and I got more points than I did on easy.
For chess, play on easy again. If you use the same set of moves each time, the computer will do the same thing as well. So, a winning pattern will always win.
UPDATE: Here is Dark Lord White's pattern:
Pawn E2 - E3
Queen D1- H5
Pawn E3 - E4
Bishop F1 - C4
Queen H5 - F7
Sudoku is easy on any level, if you want to use hints. Simply start a game, go to each open cell and hit the ? in the upper left corner. If you do that in each cell, you'll complete the level.
Finally, you need to play chess against another player. You can boost, though I found that there aren't many WP7 players who boost on TA.com, so it may take a while to get this.
UPDATE: The web game is no longer online, so you need to boost to make this easier.
Once you've beaten all four games enough times to advance a level, you will unlock the achievement.