I assume you meant you would like to run either Chromium browser or Chrome browser in one of the offered Linux public images in Google Compute Engine rather than running a Chromium OS instance. For the latter, you may check if Container-Optimized OS fits your needs per this document.
But, for running a Chromium browser, I am not sure if you can achieve that without a desktop environment on Linux. Thus, as indicated in this third party example, you may need to install a desktop environment and use a graphical desktop sharing system like VNC to access the environment and run the browser, or you could try a different approach through nested virtualization.
On the other hand, if you are just having trouble installing chrome/chromium browser (not necessarily running it in a desktop environment) and assuming you are using Ubuntu or Debian, and similar to this discussion on askubuntu, if using the dpkg command failed due to dependencies, you may need to use “sudo apt-get install -f” and retry the dpkg command again.
Furthermore, I found this third party tutorial to install a python Instagram bot on an Ubuntu instance that seems to be using both a chrome browser installation, and chromedriver, and may help you in your use-case. You may need to install the git-core package to pull the file from Github.
But as this query is more related to installing an application on a Linux machine/instance, I suggest posting your question at the corresponding stackexchange site as you have access to a larger community of enthusiasts and experts that you can share ideas with and get support from.