Find a small drill bit for metal (1/4 inch or so) and use it to pre-drill your holes. Find a couple of self-tapping screws 2 inches or so long and they should go right in.
Not very good advice here bossman..
A 1/4" drill bit is huge....Fkn huge in comparison to what you really want to use for this application....a drywall screw. Secondly you mentioned using self tapping screws which wouldn't require a drill bit to begin with.
Buddy sometimes things don't work out the way too.. you want and you need a few options..I would pick up a box of 2"drywall screws...Good to have around the house anyway...Pick up some sell tapping(bit tip) drywall screws as well...
If when you are driving the bit tips, they don't push through, take out the screw and examine the tip..Does it look like mortar or cement on the tip?
Or is the screw just dull?
If the screw is dull throw it away...Then try another...And then maybe another. Sometimes it will take you four screws...If the metal framing is heavy duty I would suggest this method before using small...and by small I mean 1/8th in drill bits because you are destined to just break the bit. So just keep driving home bit tips until you bust through the metal.
If you encounter mortar debris on the end of your bit tip screw then you need to use tapcons or plastic anchors...You know drill hole...tap plastic anchor into place then drive screw into anchor.