Car Rental Dublin, Drop off in UK?

If you want to pickup and drop off in Belfast, try the UK AutoEurope website. Autoeurope.co.uk. Select Northern Ireland as the country, and it shows all the AutoEurope options in NI, including Belfast. Be aware some rental companies charge a surcharge for going between Northern Ireland and the Republic of Ireland.

You should drop your car in Ireland or Northern Ireland, take a ferry or a flight, and pick up a second car in England or Scotland.

Another possible constraint is whether the car rental company allows their cars on ferries. I don't know about between Ireland and England but it is not allowed in certain other parts of Europe.

As Katheryne said, look at the autoeurope.co.uk website and ensure that you're clicking on the "all locations" option. I know there's a Hertz concession at the George Best City airport just north of the city. Maybe could collect one there on your way out of town since you'll neither need nor want one if you're staying in Belfast city itself.
You can't take the rental car on the ferry, so you'll have to drop in Belfast and pick up another one upon arrival in Edinburgh. For info, we did a rental from Edinburgh down to Southampton in June and the one-way drop charge wasn't too bad.

As others are saying, a one way is just either not possible like you want to do, or if they allowed, would be horrendously expensive.

I would break your trip into segments, renting a car for Ireland, maybe a second for Northern Ireland if needed, and then a third for Scotland/Northern England.

Figure you will not want a car for the major cities, so you can spend time without a car to enjoy cities, then rent for the days you need them.