Problem while compiling friggeri CV template with: ' fontspec error: "font-not-found"'

That template assumes you have obtained a license for Linotype's Neue Helvetica (€45 to €1500) and properly installed the font. If you haven't, try {Arial}, {TeX Gyre Heros}, or whatever you like, instead. Locate friggeri-cv.cls and modify the following lines (l. 53--58):

\newfontfamily\bodyfont[]{Helvetica Neue}
\newfontfamily\thinfont[]{Helvetica Neue UltraLight}
\newfontfamily\headingfont[]{Helvetica Neue Condensed Bold}

\defaultfontfeatures{Mapping=tex-text}
\setmainfont[Mapping=tex-text, Color=textcolor]{Helvetica Neue Light}

Just for completeness: It is also possible to use the simplified version of the Friggeri CV from my github account. I modified the template so that it works with a regular Texlive installation and pdflatex, without the need for luatex and biber.

(As requested.)

One alternative might be Source Sans Pro by Adobe, which gave me the following fonts:

SourceSansPro-BlackIt.otf       SourceSansPro-It.otf
SourceSansPro-Black.otf         SourceSansPro-LightIt.otf
SourceSansPro-BoldIt.otf        SourceSansPro-Light.otf
SourceSansPro-Bold.otf          SourceSansPro-Regular.otf
SourceSansPro-ExtraLightIt.otf  SourceSansPro-SemiboldIt.otf
SourceSansPro-ExtraLight.otf    SourceSansPro-Semibold.otf

Which should include enough variations to replace the Helvetica-Neue fonts. I suppose you'd want to use the -Black.otf version for Condensed Bold, -ExtraLight.oft for the Ultralight, and so on.

In terms of installing them, under GNU/Linux, I simply added them to my ~/.fonts directory. On Windows, my comment suggested this might be accomplished by moving the files to the 'Fonts' folder in the 'Control Panel' --- but that is a hazy memory from Windows XP days. (Please correct me if I'm mistaken.) Macs must have a personal fonts folder, too, but as to where that might be....

But nowadays, you can install Source Sans Pro directly from CTAN, and then use (if so inclined) via \usepackage{sourcesanspro}. This is much easier in general, but I'd be inclined in this case to make the substitutions directly.