Your idea of strapping the backpack to the rear rack has a flaw. If you hit something hard, the laptop bounces up then back down into the hard metal rear rack. It'll PROBABLY be okay in a padded case, but 'PROBABLY' eventually bites you.

Get a rear rack and a pannier bag similar to this. (Just make sure the pannier fits the laptop)

It's a great option because the weight of the laptop is on the soft canvas material of the bottom of the bag. So if you hit something and the laptop bounces inside the bag, it comes down on the flexible canvas of the bottom of the bag, not onto a hard part of the bike frame. You want a pannier that holds firm to the rack too, not one that's going to bounce outward. It needs to hold the lower rack bar tightly (most do, only really cheap ones don't) And it doesn't hur to put the laptop in a neoprene sleeve so if it doesn't boucne left-right (rather than up-down) into the side of the metal rack it's got some cushion. If that happens though it won't be a hard impact unless other things have gone wrong.