Takes about 4h if the traffic's behaving. There's good and bad.
The motorways around Manchester are busy, narrow-laned and often non-intuitive. Once you're on the M6 or M61, it's quite easy, although where these two join, and later at Lancaster, you have some nasty bottlenecks.
After that, you're in Cumbria and climb up to Shap summit. Weather changes every 5 minutes here, and it's often quite foggy and wet, but the scenery is really rather impressive.
Cross into Scotland, and you're on the A74(M), still a motorway, although narrower and often prone to lorry-going-at-52-overtaking-lorry-going-at-50 syndrome. Again, once past Lockerbie, it's hilly and pretty. But you'll need to come off here before Glasgow and head off on the A702, a single-carriageway road, to reach Edinburgh.
As you reach Edinburgh, your car will go from being freedom to quite the opposite - it's a difficult city to navigate, and a hair-tearingly frustrating one in which to park. Arrange somewhere in advance, use park and ride, whatever. But you'll not want to drive round the city. Once you're out, however...
Plenty places to step en route - you're going right through the Lake District, so you can divert via Kendal/Windermere/Ambleside/Keswick if you fancy some scenery, although the approach up here will be horrendous in the school holidays. You'll go through motorway services at Tebay, which are worth a stop in their own right - incredible views, a stream running through the café, and an excellent farm shop.
Carlisle is an interesting little city (with lots of border history and a diminutive cathedral). And then just over the border, there's always Gretna Green in all it's tacky tourist trap coach trip feeding splendour.
As an alternative, there are trains direct from the airport to Edinburgh. They only take 3h26, but the downside is they're only every two hours - otherwise, you'd need to change in Preston. You could also look and see if there's a better fare available to Glasgow - that's a dead easy journey to get to Edinburgh from there.