The roads between Belfast and Dublin are now fantastically improved compared with 10 or 15 years ago. Cheers, European Union. Back then if an old woman dropped her shopping crossing the road in a small town somewhere, the traffic was paralysed in both directions. Now it is fast smooth motorways (Interstates).
Driving around the country roads of Donegal is the upper opposite, charming and fun, but not intended to get anyplace fast.
A bit late now, but starting from BFS (Belfast International aka Aldergrove), it would be quicker to head to Portrush first, then track back to RCD before heading down the road to Dublin and cutting across to Clare.
That is still the way to go from Portrush. Head for Belfast, M1 to the turnoff for Dublin, and hammer down that road at a speed which just keeps the local police (aka Gardai) from stopping you. Speed limit plus 10% is usually pretty safe.
If you want to go down the west, which is lovely but slow, County Sligo is less of a detour.
Remember to get some UK £ and some Euro.