Is there a way I can play without having Steam running in the background?
There is no way to do so. Even if you navigate to the folder where the game is installed and run it directly, it will start the Steam client.
The closest you can get is to use the offline mode:
Offline Mode allows you to play games through Steam without reconnecting to the Steam Network every time you wish to play – this is particularly useful if you do not plan on playing over the internet and would prefer not to download new updates for your single-player games.
Please note that you must connect to the Steam Network and test each of the games you would like to use in Offline Mode at least once to set up your account and configure Offline Mode on your machine.