Breaking the chain - really not necessary
There are several ways to clean the chain without taking it apart. There are chain cleaning tools (which I really do not like, but some people do, htey have reservoirs where you put degreaser (like citrus related non toxic degreaser in) put on the chain it has a wheel and brush and this cleans your chain. Some people love this.
The way I do it is as follows:
Put the bike somewhere where I can pedal the pedals around in circles, first take some simple green and spray on the chain, take a rag and get the worst of the grease off. Pedal the pedals, hold the rag sort of still on the chain. To clean it even better, get a bunch of Q-tips, spray the chain bit by bit and run the q-tips thru each of the chain (rings) to get the gunk out. Make sure you clean the areas of the derailer where the chain goes thru etc. You can get this really well with the Q-tips. Yes, kind of time consuming, but yes, kind of cathartic (I think that is the word). Then put a tiny bit of oil on each link slowly, slowly and rub off with a clean rag using the same technique as above. This works for me. I never take my chain apart unless it has a problem. I do this on my mountain bike more often, not so often on my road bike except when it is bad weather and rainy or snowy.