If you're already PSA for the company you shouldn't have any problems getting the job (if it works like our place, at least).
Being able to effectively communicate with various departments is the biggest thing so make sure they know you're able to do so, whether it be ramp, captain, cabin crew, cleaners etc. Also concentrate on time keeping and safety / security.
Basic break-down of a charter flight dispatch:
1) Get allocated flight, start to prep paperwork (stand chit to record times, Triple AAA)
2) Request load plan from load control / do one yourself depending on how it works
3) Go out to the aircraft as necessary, whether it be a turnaround or first sector of the day. Brief the loading team leader to make sure he's happy with the load plan
4) If turnaround, bridge the aircraft or supervise pax getting off via stairs, recording the times everything happened (on chocks time, GPU connection time, airbridge/steps connected time etc etc)
5) If turnaround, tell cleaners how long they've got
6) Organise fuel
8) Monitor turn around, recording all times again
9) Brief cabin crew on expected pax number, specials etc
10) Pester to cabin crew to stop gossiping and start boarding pax
11) Collate necessary paperwork for the crew. Pax manifests, specials lists, meals lists, loadsheet etc
12) Confirm pax figure with gate staff after boarding
13) Confirm bag figure / loading with ramp team leader (most important part of turnaround arguably)
14) Pop into the pointy end, grab your loadsheet and say cheerio
That is of course a very simple break down of it. Add in your random drunk/wanted/ill pax, late service partners, bad weather, rude/arrogant captains and inevitable delays.
Also invest in a bendy dispatchers watch, best thing I've ever bought
"on time, every time".
Good luck, PM me if you've got any questions.