It's difficult to get an exact match of your client's resolution because of a couple of issues.
DFDatabaseAdmin is correct that just because you are targeting 1920x1080, doesn't mean that's the resolution that's going to be available. The scrollbar, borders of the window and any addons that take space on the browser's viewable area of the website (the "viewport" or "screen"), then you actually have less than 1920x1080.
A better solution for responsive is to always account for 15-20px of space for each media query, so that the scrollbar and window borders don't prevent that magic width from being reached. So if I want to target 1024px width and lower, I would specify 1010px as teh max-width.
As for your client's PC, you could do a media query on his min-width, rather than max. So instead of this:
@media screen and (max-width: 1920px){
#code here
}
You could use this:
@media screen and (min-width: 1900px){
#code here
}
When I build responsive websites, I usually use foundation from Zurb as my responsive framework to save a bunch of time, but if you're wanting a full custom experience, then coding responsive by hand is the way to do. For my responsive websites, I use 1010px (desktop), 780px (tablet), and 600px (mobile). However, don't be surprised if you may need to create your own side queries to change certain areas earlier or later though. Hope this helps.