This is a good plan and it is entirely doable. Anyway, there are certain specific things that you have to take care of and if you feel still confused, just write it here.
Well, to reach Mljet, especially its most precious part, Mljet national park, you have to take a catamaran that sails daily from Dubrovnik at 09:15 and arrives in place called Polače (this is basically the entrance of the national park) at 10:50. Here is the timetable for June: http://www.gv-line.hr/raspprint.php?linija=3&red=62 .
Now, to reach Korčula from Mljet without returning to Dubrovnik, you can count on twice daily service provided by the company Jadrolinija. Here is the timetable, but in croatian language: http://www.jadrolinija.hr/pdfs/DUZOBALNA%202013.pdf . Well, the ferry departs from Mljet, but from the port called Sobra, every Thursday and Sunday at 10:40 and arrives in Korčula (town) at 12:40.
In order to reach Sobra on time, please, just use the local bus that departs from the small village Pomena and passes through Polače in the morning, so that the passengers can reach the ferry before departure. The bus timetable changes from season to season, so just inform yourselves at the national park reception.
It is a pity that the catamaran does not proceed all the way to Korčula, but it is the case only during the peak season i.e. during the months of July and August. So, you can make your island hopping using this ferry line.
From Korula it is very easy to reach Hvar (town). There is daily direct catamarn that departs at 06:00 (I know, a bit too early in the morning) with arrival in Hvar at 07:30. Here is the timetable: http://www.krilo.hr/en/sailing_schedule.html .
After that, you have a daily direct catamaran service from Jelsa on Hvar island to Bol on Brač island. It departs at 06:00 (Sundays at 07:00) with arrival in Bol at 06:25 (07:25 Sundays). Here is the timetable: http://www.jadrolinija.hr/pdfs/9603.ljeto.2013.e.pdf . To reach Jelsa from Hvar you will need to use the local bus, but you will have to sleep the night before departure at Jelsa because there is no such an early bus from Hvar to Jelsa in order to reach catamaran departure. Here is the bus timetable: http://www.tzhvar.hr/UserFiles/File/pdf/info/Bus-Time-table-2012-2013.pdf .
From Bol, just take a catamaran to Split and return by bus back to Dubrovnik.
As far as Vis is concerned it is important to know, that it is not connected to Brač at all and with Hvar there is just one weekly catamaran line on Tuesdays at 16:25. So, to reach Vis you have to plan to use the catamarn on Tuesdays or go first to Split and from Split to Vis. From Split to Vis, the ferry and catamaran connections are just fine.