I personally don’t worry much about a little grime, and just wipe it with a good cloth or toothbrush to get the loose stuff out. Over time and use, a tool is naturally going to collect a little oil and dust. As long as it functions ok, I’m good.
But if I feel it must be cleaned thoroughly, I might run it under hot water and scrub. Then I would use a heat gun (preferred) or put it in the oven at high heat to force the water away. Then thoroughly oil, including a drop in all screw holes. Good chance of flash rust though.