There are a couple of options.
See if there is a CBS app in the LG webstore. I do not believe there is, but I have not checked recently. There is CBS All Access, which is a pay service. It has some great content, new and old, for just a few bucks per month, and will have a lot more than is available on their website.
Get a Chromecast. This would be my number one suggestion. With it, you can use the CBS app on your phone/tablet to cast right to the TV, or from a Chrome browser on your desktop. the Chromecast does all the work, with the mobile device just acting as a remote, so you can continue to use it while casting. There are other devices, like Roku and Amazon Fire, but I am not familiar with their specific app or casting support. I can tell you that, last I checked, Chromecast is the only device that supports screen-mirroring from both iOS and Android devices.