This error from npm is because you don't have node-gyp configured so npm doesn't know how to compile the binaries required for the particle-cli.
One way to fix this is documented here: GitHub - nodejs/node-gyp: Node.js native addon build tool
Install all the required tools and configurations using Microsoft's windows-build-tools using
npm install --global --production windows-build-toolsfrom an elevated PowerShell or CMD.exe (run as Administrator).