Ah, my least favourite part of a detail
Lots of approaches, mainly common elements though. This is what I do. I'm not a pro, just a keen home/amateur detailer.
(There's probably threads on this already, and likely a detailing board somewhere, but my thoughts)
All at your own risk
Cold engine
Choose your location wisely, as it will spatter.
Bag any open induction cone/s or breathers
Glove up and take any other PPE precautions required. Goggles and wellies and a scene of crime type hazmat suit are useful in some cases. Seriously, pressure washing and engine bay at close quarters can fire all sorts of stuff back at you.
Read labels of any products you're going to use.
Remove loose stuff - vacuum, compressed air, use hands for leaves etc.
Some people opt to tape up exposed electricals so just assess whether you might be more comfortable taping/bagging any exposed wires or aftermarket electrics that aren't sealed very well (I've never done this but situation dependent)
You can then rinse off the entire bay at this point, depending on how much dusty crud there is and how tough any hardened gunk looks. Water can be your friend but will/can also dilute effectiveness of any products you then use, so all dependent on your sitch. No great shakes and I'd say it is probably more beneficial in most cases to wet it and blast off loose dirt so you have less work to do manually.
So loose stuff removed, electrical assessment done, and you've made it all moist by now
Have a brew
Apply gunk remover, degreaser or APC over everything, taking care not to get anything too aggressive on outer panels. Nowadays I just use varying solution strengths of Bilt Hamber Surfex HD as my all purpose cleaner through to heavy degreaser, but depends what you have in the cupboard. Pretty sure I've used CIF and Flash before
Let it dwell to take effect - 5-10 mins maybe?
Agitate with a stiff brush, tooth brush, old microfibre
The bonnet sound deadening can get super mucky and is quite resistant to becoming clean, just douse with cleaner then drive the dirt/cleaner down from the highest point (front of bonnet as it's up in the air) down or out.
Pressure wash/rinse it all.
Assess your work and revisit the above steps as required as you generally move detritus from one place to another on the first pass
Let it dry off and use towels you're not too fond of. Or blue roll, but think of the penguins.
At this point you have options as to finish. Shiny stuff can be applied, silicone spray can be used to rejuvenate hoses etc and plastics, or you can use vinyl detailer type stuff like the meguiars stuff. Personally, as with the whole vehicle, I don't like to go for the greasy second hand car sales pitch level of finish. Clean and a bit of a sheen for me
Exposed paint inside the bonnet and wings can even be polished/waxed if you like. Waxing makes it easier next time... Just saying.
The world is your lobster!
Engine covers do look nicer and tidier and make future cleans simpler
Oh, and we need pics!!!
Good luck