How do you get your Steam games to run on Ubuntu through Wine or something similar?
There is no best way to do it, as such. Both Steam and Wine are ever-changing, and today's best method might break tomorrow*.
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People have reported that Steam works with just standard Wine.
Install Wine from the Software Center (or click the shopping bag icon).
Right click the installer executable, go to Properties → Permissions and check Allow executing file as program, then right click it again and select Open with Wine Windows program loader.
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If it turns out that doesn't work for you, you can try using PlayOnLinux.
Marco has detailed the instructions on how to get it working behind the link.
If it still doesn't work, you can give Crossover Games a go. Though it is number three on this list on purpose.
In general, it seems that Steam will work okay. You should try to run it using out-of-the-box Wine first. If you run into any trouble, feel free to ask a question under the wine tag at Ask Ubuntu.
Also see this site for Steam games that work with Wine (and, by extension, PlayOnLinux).
* Note: If you get it to work, updates to Wine will most probably never break it. But it may well be the Steam people are fighting off Wine compatibility. It may also be they are actively trying to make their program run well under Wine (games publishers have been known to do that).
In addition to Wine, WineTricks helps really much. It installs all the required drivers, libraries and frameworks into Wine.
Steam Games on Linux website contains Wine configuration and things you need to install (including WineTricks for some games) to play different Steam games.
I successfully played Alien Swarm and Half-Life 2 Deathmatch, though they don't seem to need any special configuration.
As of 15 Feb 2013, Steam is officially available on Linux!