I am looking for a process in ArcGIS that takes a point feature class and groups the points into a user-provided number of clusters.
k-means clustering is exactly what you want. Version 10.1 has k-means clustering... if you are beta program eligible you could try that. You could also try doing your k-means clustering in R and then bringing the results over the arcgis. I think there are also some 3rd party scripts out there for k-means clustering using versions 9.3 and 10.0 too.