YES!! of course you should.
Opinions vary on what is the best way to clean your chain, and what will or will not hurt it.
I've heard everything from soup to nuts for cleaning chains. (okay maybe not nuts but you get the point).
I use MOTUL chain cleaner... IMO its the best, with the least amount of effort. Put your rear tire up on the stand, spray the chain while rotating the rear tire.... then once it's all coated in cleaner, hold a rag under the chain and rotate the tire again. You won't believe how grimy and gummy your chain gets.
Once that's done, find a really good chain wax, or lube (again, I use MOTUL) and apply to the chain while rotating the rear tire. Even with good wax or lube, you need to let it set up. Don't spray your chain and instantly go peeling out... most of what you put on will fly off and end up on the inside of the chain guard, the undertail, etc.
I had an ex-girlfriend that said something to me once and it stuck with me... seems appropriate in this situation as well... she said "You can never have too much lube". I thought she was talking about greasing the chain on your bike... now I'm not so sure.
