Sure you can, and should. It's not at all hard, and you don't need much in the way of tools.
First thing to do is blow all the loose junk off, with a shop vac hose hooked up to the exhaust. This reduces the amount of dirt that will turn to mud.
Then, with a cool engine, spray all the metal and plastic pieces down with some all purpose cleaner - simple green, 409, something fancy from the detail aisle... then scrub it with a kind of firm parts cleaning brush....it lets you get down into small spaces. Rinse, repeat as needed.
If you want to be fancy, spray and wipe down the plastic pieces with some protectant. I like 303. Get the hood gaskets too. Then I lubricate the hinges and lock with silicone spray.
There are a ton of youtube videos on detailing cars, some include doing the engine.