How to add non-Steam games to Steam's Library?
I downloaded a game from a website other than Steam. How can I link the game up with my Steam account so that I will be able to get the badges and achievements that come with the Steam version of the game?
Unless the game is designed to work with Steam via the SteamAPI you will not be able to get achievements or badges with that game.
Generally speaking, if you're not buying a game on Steam it will not be written for the SteamAPI.
If you purchase a Steam key elsewhere then you can 'Activate' that key on Steam. You would then get that game in your steam account just as if you'd bought it through Steam — a copy that was written with the SteamAPI and thus has badges and achievements.
Non-Steam games cannot link with Steam badges, achievements, etc.
But you can add them to your library and be able to launch then from steam, and have the in game overlay (shift+tab) in game.
Go to your steam library and click add a game.
Then: Add a non Steam game...
It will now start to list all of the possible things that can be added.
If it does not find the game you want, you can simply press Browse... and find the .exe file of the game you would like to add.
I hope this helps!
Sadly if you add a non steam game its achievements cannot be added because the game has no affiliation with Steam. The game would have to be sold and affiliated with Steam and its API to have achievements.