I think a week is too short for what you're proposing. You could do it, but you're not going to have to time see much. Have you plotted out the travel on Via Michelin? Its three hours to Aberdeen from Edinburgh with no stops. And that's without the 25% add on time. And that's going up the coast and not through the Castle Country.
The way I would want to go would be, after a day in Edinburgh, I'd drive up toward Balmoral and stay in Ballater. I wouldn't do the coast. It's a scenic drive, there a couple of ways you could do it. One way would be to go via Dundee and stop off to see Glamis Castle, the original home of the late Queen Mum. Or if you would rather get out and do a little walking go via Dunkeld and walk in Birnam Wood or to the Hermitage. Have lunch at the Taybank. BTW either way as you pass Balmoral, and instead of heading east immediately, turn left and head immediately up hill to a wonderful gallery. This will be a long day.
Day three go to one of the many castles in the area. Castle Fraser would be a good choice, but there are many others. Then head east and spend the night in Inverness.
Day four If you are interested in history, go out see Culloden Field just outside of town. Spend the day driving to Isle of Skye. It's a beautiful drive. You'll want to stop by Loch Ness, Castle Urquhart, and the glens and lochs of the drive west. If you want to stay in Portree, bear in mind that it will take you about an hour once you're on the island to get to Portree.
Day five if you want to see a bit of Skye, then you can't get all the way to Oban, so I would say explore Skye and then catch the ferry to Mallaig.
Day six, you're off to Glasgow via Glencoe. It's a long driving day, but it will be very scenic. The drive from Mallaig to Fort William is absolutely wonderful.
Day seven is Glasgow. I hope you are leaving the next day. :)
It's not a trip I would do as you are changing hotels every night. A shorter trip might be to spend a couple of nights in Edinburgh, drive up to inverness spend the night. (There are some things you could see on the way up.) Drive to Portree the next day checking out Loch Ness, and spend a couple of nights on Skye. Then take your time driving back to Glasgow either stopping somewhere on the way down or planning on two nights in Glasgow.
Others will no doubt have more suggestions.