To set your profile so that Chrome uses it, assign it to your monitor system-wide (how to do this depends on the OS but DisplayCAL can take care of it), and make sure that chrome://flags/#force-color-profile is set to “Default”.
Then, to see how far off Chrome is (at least within sRGB), you could use DisplayCAL’s verification tab with “Display” set to “Web @ localhost”, “Simulation profile” set to sRGB, and “Use simulation profile as display profile” checked.
(With Chrome’s color management enabled, DisplayCAL can only test how Chrome displays colors within sRGB because it sends its colors via CSS, which only supports sRGB colors until CSS Color Module Level 4 gets more traction. If the color profile in Chrome were set to “sRGB”, then it would pass the CSS colors to the screen unchanged, allowing DisplayCAL to measure the monitor’s whole gamut, but that’s not the point here: the point is specifically to verify how Chrome translates sRGB values into actual colors.)