Aachen is known for its bad weather in Germany. Supposedly, it rains all the time. Personally, I think it is true and it does rain a lot; however, that's probably a general German phenomenon, not limited to Aachen. You should definitely get an umbrella! (The drugstore dm is good for inexpensive and sturdy umbrellas.)

Winter

Lousberg in Winter

In winter it gets really cold. Starting around November, temperatures will often fall below 0°C, though this can vary from year to year. The coldest months are usually December through February. Temperatures can easily fall below -8°C,  and on really cold days it can be even colder. Don't expect white Christmas. Yes, it does snow in Aachen, but not necessarily before Christmas and some years it won't snow at all.

Summer

Summer usually isn't that hot in Aachen (or in Germany in general). The hottest temperatures I have experienced have been around 37°C, but temperatures above 30°C are already considered to be hot in Germany.

Miscellaneous

If you're not used to it: Be aware that ice can make bicycling extremely dangerous. Sometimes your brakes will not work at all because it's too slippery. That is also the case for rainy days or even worse when wet leaves are on the ground. Be cautious when riding your bike!