Eagle Cad: How to get copper on both layers for a pin header
Since you did not mention how many Layers your PCB is going to have, we have two scenarios.

If you got only a 1 layer PCB, then the best solution is to put the SMD parts on one side and the Pins on the other, so you can easily solder them to the pads.
Side view on PCB:


This is obviously the best option as long as the parts on the other side are almost of the same height. Depending on how the PCB is going to be used, you might want to add legs.

Since your picture just shows the layers top (red) and pads (green), but no bottom layer (blue) make sure its checked and therefor visible (you might want to turn of the top layer, when routing the bottom layer).

Afterwards you should check whether you got the right library for the Pin Headers. There are definetly ones with pads on both sides of the PCB, for instace in the standard library "pinhead".

If you place a via in eagle (2 layer PCB) you always have connections on both sides.
Make sure to check the "via" option if the vendor has one, because they are not always added by default.

EDIT:

I tested it in eagle and there is a connection between top and bottom layer of the PCB over the pinhead:



So if you order correctly, then there should be a connection between both layers over the pads on both sides. Just use the rule checks from eagle to make sure everything is connected (run ratsnest and turn off all layers except "Unrouted").