SP: Level 60 is the highest level you can obtain in Mass Effect 3. Once you hit this level, you will automatically gain the achievement. If you played Mass Effect 2 and imported your character, you will start the game leveled up. You can be as high as level 30 when you start your new game, meaning you are already halfway there.
MP: You must stick to one character and focus on leveling it up. It doesn't matter if you win or lose the matches, you will still gain experience. This is probably the faster option of the two to unlock this achievement.