Best weed killing/ grass killing product

I have an area in my backyard that is grass/weeds, but quickly overgrows. It’s a pain in the arse to maintain, and my neighbor does me no favors with his out of control backyard. I’ve also had more bugs than normal, and the occasional rodent that my dogs kill.

Long story short, I want this area under control. When it’s dead, I want to take it off and maybe just turn it into a rock garden, or even lay paving stones. I do not want vegetation.

What’s the best way to go scorched earth on this shite?