We automatically update the Bark Desktop App on Windows.
In the event it's not updated automatically, follow these steps:
Open the Bark Desktop App from your applications list or from your desktop.
Click on the arrow ^ near the time on your taskbar.
Click on Bark.
Click on Check for Updates. It will then install the update if applicable.
Alternate Method
Download the latest version from bark.us/ios.
Open the installation file and follow the on-screen prompts.