How many points do you have? Do you have access to some good elevation data?
One options is to use the Google Elevation API [(free) API key suggested] if you don't have [or, like me, can be too lazy to get] quick access to a DEM or other good elevation data. Note that the DEM resolution will most likely be less than what you can download, but if you are just trying to see how far off your GPS is that might be fine. It will give you the lat/long as well as the elevation (meters) and the resolution of the DEM, which is nice.
If you need a point:
If you are looking at a path or a slew of points in succession: https://developers.google.com/maps/documentation/elevation/#Paths
I once compared my sub-meter horizontal GPS handheld data with some LIDAR data and had huge differences in some areas, so hopefully you're not using the GPS vertical for anything too accurate.
While the link above works fine for me, Google encourages you to use the API for access - more info below