Am I responsible for maintaining the alley behind my house?

Yes. Typically, the property owners on each side of an alley are responsible to maintain to the center of the alley for the width of each respective property. This includes  grass, weeds, trash, etc., that may be found in the alley, regardless of how it ended up there. Weeds on the other side of your fence along the alley are still part of your property, and therefore your responsibility to trim and/or spray.