Playing SC2 Replays with double click

It may be that the replay is from a previous patch, which can cause this issue.

You can view the replay by placing it in the replays folder located here:
%username%\Documents\StarCraft II\Accounts\xxxxxxxx\x-xx-x-xxxxxxx\Replays\Multiplayer
There should be a shortcut in the StarCraft II folder pointing to your account.
You can then access the replay in-game.

StarCraft II will load into the previous patch and play the replay. Note that you will be offline on Battle.net while viewing an old replay.

You cannot launch replays from outside of Starcraft 2 if it is already running. If you close SC2 completely (not just minimize) and then double-click the replay, SC2 should launch and start the replay after you log in but if you then try to do the same with another replay while SC2 is still running, it will just bring SC2 to the front without actually doing something.

As Ampersand noted in his answer, you can put replays into the SC2 replay folder

%username%\Documents\StarCraft II\Accounts\xxxxxxxx\x-xx-x-xxxxxxx\Replays\Multiplayer

if you want to watch multiple replays in a row. If i remember correctly, there is also a button on the replay screen that opens the replay folder in the windows explorer if you can't find the correct folder.