I turn mine over, thunk it a few times in pan or bucket, and 95% of the hop matter falls right out. Then I just put it in the utility sink and spray it down--first with it right-side up to knock stuff out of the bottom (it's recessed, so hot break tends to collect there a bit while boiling), and then I flip it over and spray it from the bottom. I don't bother to clean it with chemicals; the force of the water (I use a spray nozzle in the sink) cleans it out well enough. Takes me no more than a minute.
I basically treat it like my immersion chiller: I want it cleaned off enough that there aren't gobs of boil detritus left on it, but I don't care if it's sparkling, since it's going back in the boil and will be pretty well sanitized. I've used it for 6-7 batches, and it looks used, but functions like a champ.
You could probably use a dishwasher if you wanted, but I'd be certain to get all, or at least most, of the hop matter out first. I don't dry mine; I just spray it with the hottest water I can, smack it a few times to release most of the water held on by surface tension, and leave to air dry in the garage by hanging it from its integrated hook. It's 100% stainless steel, so it should be pretty resistant to rusting.