The units of the graph and how to change them depend on which axis you're looking at.
First, the horizontal units are the same as your current map projection. If you are using a Geographic coordinate system, the units will be degrees. If you're using a standard UTM projection, the units will be meters. The only way to change this use a different projection, which has the desired units. Note that you can create a custom projection by starting with the correct UTM projection and then editing it to change the unit of linear measurement and saving it as a copy (say, by tacking 'feet' to the end of the default name). You may or may not have to actually reproject your data as opposed to just changing the project of the data frame and reprojecting on the fly.
The vertical units are controlled by the units used in your DEM. If your DEM is meters and you want feet, you'll need to create a new DEM by running Raster Calculator on it and multiplying the values by 0.304800609601219 (or dividing by 3.28084), then use the new DEM for your Profiles.
Related question: Problem with interpolating a vertical profile in 3D Analyst (ArcMap 10.1)