A few things....my opinion is that computers generally don't need to be cleaned as much as people think they do. I base this on 30 years experience running many PCs in a manufacturing plant.
That doesn't mean you can't clean it if you want to. I clean mine sometimes just so they look nicer.
If you just want to clean it up a bit.....remove the side panel or whatever panel and use compressed air to spray out the dust. Stay far enough away from the components that you don't damage them.
As far as the wrist strap....just touch the case before you start and touch it every so often during the cleaning.
Don't touch any of the components.