I'd echo what Lowfare is saying. Some of the most beautiful scenery in Ireland is to be found in the North. The Antrim coast, north of Larne all the way up to Ballycastle and further to the Giant's causeway as well as the Mourne Mountains in county Down are well worth visiting. IMHO, the city of Belfast does not have that much in terms of sights as such but it is a great place for restaurants, pubs and clubs.
One thing to bear in mind, though, is that July is the peak of the loyalist marching season. You would not be in Ireland for the Battle of the Boyne anniversary parades (July 10-12th). However,you'd definitely want to steer clear of Portadown/Drumcree on July 2nd-9th (but then again, which tourist would want to go to Portadown at any time of the year, in any event?). If you need info on when/where parades are planned, you can look at the Parades Commission site at http://www.paradescommission.org.