The latest Steam Beta update adds Dualshock 4 support, after surprising us with the announcement at Steam Dev Days last month.
The native Dualshock 4 support is now available in the most recent Beta update, where you can get the client, and you will no longer need third-party software.
The reason the Dualshock 4 was chosen was because of how similar it is to the Steam controller, and therefore the native API support means you can map the trackpad, gyros, plus buttons to keyboard, mouse, or x-input outputs. The controller also works with Big Picture.
We do know that there will be other controller support coming in the future, but for the time being, Steam is sticking with the Dualshock 4 due to its similarities with the Steam controller.
You can get the update now opting-into the beta from Steam under the Account section.