While the documentation at the official Blazor site is great, I wasn't able to build the solution without a few extra steps. Importantly, the version of Node doesn't support the latest version (11.x+). I recommend 8.12.0.
Finally, if you are trying to build in an environment in which the Blazor Language Service VSIX package is already installed, you're going to have trouble building the solution as it's going to complain that the version of the package you're building is not greater than the version of the extension you've already installed (presumably because you've installed that extension to enable Blazor development locally (as in, you are consuming and not developing the Blazor ecosystem). The cleanest way to combat this problem is to create a VM for the purposes of your Blazor pull requests.