When we have guests here we usually do the following for a five day itinerary:
1. The center of Split, climb up Marjan, museums (there are quite a few nice ones including the Mestrovic Gallery, various archaeological museums, the basement of Diocletian's Palace, etc.), climb to the top of the church tower in the center of the palace, dinner in town.
2. Krka. A real gem, with an upper and a lower section and a monastery on an island in the lake in between--you can spend the whole day here or add Sibenik.
3. Solin and the Roman ruins there, Omis (south of Split) and the Centina River valley (rafting is possible in summer), Kastela & Trogir (north of Split) and Primosten.
4. Hvar (a two hour ferry ride or a slightly shorter catamaran ride).
5. Brac (Bol)--another ferry. or alternatively, Plitvice Lakes (that's about three hours by car). There are other national parks nearby like Paklenica or Mljet but Plitivice definitely deserves it's Unesco World Heritage status.
And that doesn't even include Zadar to the north or Brela and Makarska and Mostar to the south.

So five days is a lot but you can go further afield and find some really nice things to do. There's a lot of fun hikes in the nearby mountains with wonderful views of the coast and the islands, too. Dubrovnik is fantastic and if you're there longer you can also include Hercegnovi and Kotor in Montenegro (just make sure your car insurance is valid for that leg).

Have fun!