To be explicit: you can run the Cloud SQL Proxy directly from your PC. No need to use a Compute Engine VM! Please feel free to post back if you are having trouble setting up the Proxy, we'll be glad to help out.
In general, there are three major steps:
1) Download the proxy: on the
page previously linked, you can download a precompiled version for your OS/CPU combination of your system.
2) Run the proxy: there are a few examples at the bottom of the page on how to start running the proxy
3) Connect to the proxy: depending on what flag options you passed to the proxy when starting it, you will need to configure your application (mysql command line, for example) to connect via TCP or UNIX socket.
If you have
gcloud installed and have run `gcloud auth login`, the latest version of the proxy will just use those credentials when accessing Cloud SQL Databases.