You'd check in for both flights from home. You'd have both BP's in your possession before you board the Frontier flight.

Which airport are you talking about? That would help us give more specific information. In some airports, it's a simple walk from one gate to the next. In others, it's a terminal change and potentially a 2nd pass through a TSA checkpoint.

I agree with the others about how a 1-hour turnaround is a bit risky, even for a same-terminal connection. Since this doesn't smell like a Mileage Run, I'm guessing you have some sort of real purpose in the destination airport? If your intent is to meet someone there, they may not be able to enter the secured area. (There are ways, and there are threads here dedicated to how to do it, I'll leave it to you to search if applicable.)

I'm also assuming that with these two particular airlines this is entirely within the U.S., right? If not, then all bets are off...don't even think about the 1-hour turnaround.