I've cleaned/lubed the bushings quite a few times over the decades. The nice thing about doing it these days is the quick-detach link that is on many new chains, so using a chain breaker is not always needed. It's a small item that makes drive train service soo much easier.
You don't need to break the chain to clean the pulley(s). If you want to remove it, just take the chain off the crank so it is slack, and then undo the small allen bolt that holds the pulley in place.

Unless they are really dirty, you can usually get away with just squirting some aerosol type lube right at the heart of the pulley. A few miles of riding will allow the lube to work its way into the bearing and silence it, assuming the sound you hear is a "squeak" and not a "crunch." Once it is quiet, just wipe the excess lube on the outer surface away, so it doesn't attract more dirt.