Hi steve, once you are on the dual carriage way through the Boro you will see the sign post for the A19 North and South. It is important that you keep to the left lane after the town centre because this is to get into the turn off road that goes north. Once on the turn off then go into the Right lane and then onto the A19 proper.
As you go up the A19 you will miss the first turn off which is for Billingham, stay in the center lane. You will take the next turn off a few miles up the road. I think it has "BISHOP AUCKLAND" and it. Stay on that (about 10 miles) till you see the A1m and go north. Then go up that for TWO junctions then you will see the sign for Durham city. Dont worry if you see a sign for Durham services on the first junction, that is NOT Durham.
Once you have turned off for Durham itself then that road, a dual carrigway goes down into Durham City itself. I dont know where you are going to stay though so you will have to let us know or you should be ok from there.