Out of the two Nitrogen atoms of the 4-(Dimethylamino)pyridine , the pyridyl nitrogen is susceptible for alkylation as it is less sterically hindered than 4-Dimethylamino in pyridine.

The one in the ring

Both N atoms are sp2 hybridized. Although the exocyclic N seems to be more basic (but is not) is less nucleophylic. Then alkylation occurs on the ring N atom (by kinetic control), as would also protonation occur (by thermodynamic control).

Is there any way to get both alkylated ?

You can try with an excess of alkylating agent, but it is not very likely, since you would get a di-cation!