Durham Tees Valley Airport is badly named because it's not very close to Durham. (It's 25 miles away). You need to get to Darlington, which is only 6 miles from the airport.
National Express can get you to Darlington but it's a lengthy journey. The first departure from Edinburgh isn't until 0930, taking 5 hours 10 minutes to get to Darlington (with no changes), arriving at 1440. The fare is £32.50.
It's also possible to leave Edinburgh at 1230, for a 4 hour 20 minute journey to Darlington, arriving at 1650. (Same fare but you have to change at Newcastle).
From Darlington (Stand K on Tubwell Road) Arriva bus service 12 (NOT 74, as stated on the airport website) runs at 20 minutes past each hour, with a 27 minute journey time to the airport terminal.
However it would probably be far more convenient to get between Edinburgh and Darlington by train. There are frequent services, with a journey time of around 2 hours. The fare on Cross Country services is £33.00 but East Coast trains are cheaper at £27.00:
Chris