Can I save my campaign on computer 1, and then continue on computer 2?
Is it possible to somehow save your campaign progress in StarCraft 2: HotS to a cloud, so I can access it from multiple computers?
Yes and no. Your progress is stored in the cloud, but your actual savegames are not. You can fix this by simply copying your savegames between your computers. You can do this by hand or use a synchronization program. I'd recommend Cubby or Teamdrive, because both offer you to sync any directory on your PC(s) for free. More popular Solutions such as Dropbox are possible as well, but you'd need some workarounds, because you can't just sync any random folder.
Your savegames for the game can be found under [My Documents]\Starcraft II\Accounts.... For Windows Vista and above, this usually resolves to C:\Users[Your Windows Account]\Documents\Starcraft II\Accounts.... The exact name of the Folder depends on your realm as well as your account ID (don't worry, usually you should only have one option/sub folder anyway).