Your road-trip clockwise around the island sounds fabulous!
I had a look and while companies usually allow to travel between the Republic of Ireland and Northern Ireland, I couldn't really find a rental agency that allows one-way rental from Dublin to Belfast. Entreprise
for instance allows one-way rentals for an additional fee, but I would check with them first whether it also applies between Dublin and Belfast. Their online search tool doesn't show any results for this option.
What I would suggest instead is dropping off the car at Dublin Airport and take a bus to Belfast from there. (plus the money you would save on the one-way fee may offset the cost of the bus tickets). The Transport for Ireland
journey planner is quite useful here.
Any idea what you'd like to see and do during those 8 days with us?