Hi,
Sorry to hear you're experiencing problems getting this working, this workflow should work ok. If you're using a mapped network drive, you'll need the same named drive on both machines (e.g. V:\Data\... ). Otherwise, if you want to edit from the client machine to the other server machine you'll need to use a UNC path (e.g. \\SERVER\C\Data\...).
In ArcMap you will need to add the data from the mapped network drive / virtual drive (I use the doc command subst to create a "virtual drive") so that when you create the map package and choose the "Reference all data" option it will persist the reference to the data via your V:\Data.. location. You should find the MPK that gets created is very small (probably <50KB).
Then when the ArcGIS Runtime starts the LocalFeatureService it will unpack the MPK to your user profile (e.g. C:\Users\mbranscomb\Documents\ArcGIS\Packages\<package_name>\). In that location there will be a v101 (i.e. 10.1) folder which contains the just a couple of map documents (and MSD and an MXD) which the ArcGIS Runtime is using to create the service. You should not see a <map_name>.gdb Geodatabase or any other data. This means that the MPK was correctly created referencing the data by the original path (e.g. V:\Data\...).
Starting a LocalFeatureService from this MPK should now be referencing the data in it's original location via the mapped network drive path. I've tested via both mapped network drives and virtual drives (the subst command approach) and in both cases the internal path within the package is correctly pointing to for example X: or Z: rather than C:.
Cheers
Mike