Oh dear. If you have checked bags, you're going to need a minor miracle to pull this off. Is there any way you can do this trip with hand luggage only?
But assuming no... since your flights are on separate tickets, it is unlikely that Qatar Airways will be willing to check your bags through to Doha. (You can and should ask at the counter when checking in, though, since your 80 minute connection is just barely above the legal 75 minute minimum in Dubai.)
This means that at Dubai, you will have to go through Immigration, pick your bags, go through Customs, go to Departures, check in for your flight and drop your bags. Since the Qatar Airways checkin in Dubai closes one hour before departure, it will be virtually impossible to do all that in twenty minutes; the airport is so large it can take you 20 minutes just to walk from the arrival gate to Immigration. Your only hope is thus to check-in online (available up to 36 hours before departure) and drop your bags at the fast drop counter (which usually closes around 30 min before departure; can't find a figure online though?). But even this will be seriously pushing it, and you're screwed if eg. your incoming flight is delayed, since you're not going to get any refunds if you miss a flight you've checked in for.
If you do miss your onward flight to Doha, and frankly that seems quite likely, you'll be at the mercy of the airline. If your ticket was cheap like you imply, it's unlikely they'll permit you to switch to another flight, although you might get some credit towards a new, much more expensive ticket.