When you conquer an enemy, a pop-up will appear above your name at the lower left listing the amount of Credits, Carriers, and Territories you received from that conquer.
This achievement can only be obtained in the custom gametype, against skilled or expert opponents. The CPU you conquer also has to have some credits on hand as well as a carrier.
The territory part of this can be tricky. If you take a carrier over and land on a CPU's last territory, this will not count as territory gained for you (the pop-up will list a 0 under territory).
What worked for me was attacking a CPU that had an island of three spaces - two had walkers on it, and the third was empty. In the same turn I used two carriers and took over both of his territories with walkers on them. When this killed him, I received his credits, his carrier, and most importantly his last unoccupied territory. The achievement popped at that point.
You may find this easier to achieve in a 4 player match as the CPUs will fight each other and open up possibilities for a attack like the one I made, however this is not strictly necessary.