Assuming the opponents aren't good, you can probably win by just using Mr. Game & Watch's back aerial a lot. It's really funny how bad players just fall to pieces completely unable to answer it.
Also, in terms of what not to do, NEVER use Judgment Hammer, and use Chef very rarely. Judgment Hammer could always be replaced with a better, reliable move; the horrible range and speed don't mix well with the honestly really low chance of the hit doing anything but sucking. Chef is extremely punishable; don't try to use it to approach or when they are in any situation to approach you as it will just get you hit. An example of a good time to use it would be to pressure them from the main platform of Green Greens when they are on the far platform. A good example of a bad time to use it would be on Final Destination when you both are on the stage (it's a mediocre ledgeguard too so you probably just won't use it at all on that stage).
I don't know what stage rules your school is using. The best fair stages for Mr. Game & Watch are Green Greens, Norfair, and Port Town Aero Dive in my experience, but unfair stages could help you quite a bit. If they allow Skyworld, counterpick it against guys like Olimar with poor recovery and laugh as your down throw spikes through clouds. If they allow Pirate Ship, spend as much of the match as possible in the water and try to abuse the invincibility of the up special and their limited options in handling the key in the water. Mr. Game & Watch doesn't have much he needs to avoid stage-wise; Mario Circuit is his only actually significantly bad stage that has any reasonable chance of being allowed.
Speaking of the down throw, down throw to down smash is a guaranteed hit against much of the cast if your opponent sucks and doesn't tech the down throw. If they are a heavier character or don't suck, you might have to use some tech chasing out of that; up smash is good if you can get away with it, but players very familiar with it will not be hit and maybe punish you. In that case, dash attack, down tilt, and just running up and stopping to punish the spotdodge are going to be what you need to do.