As Dan Patterson explains, there are tools that combine raster and vector formats. Especially the tools you want does this. You can even use the ASCII raster as input. However, does this file have a .prj file too? The .prj contains the projection information which ArcMap uses to locate the coordinates on the globe. Make sure you work with the same coordinate system for both your counties featureclass as the raster (.asc).
If you don't have a .prj file you will first want to use the Define Projection tool to assign the appropriate coordinate system for it. In case it is different from the counties you want to project the raster to the coordinate system of the counties.
From the tools Dan Patterson suggested I think the Zonal Statistics As Table is the most appropriate tool. It requires the Spatial Analyst extension. It will create a table that you can join back to the counties featureclass.