In order to promote a Commander to max, you will need a bunch of Commander xp, and a few of promotion orders.
The xp is gained by using ANY Commander in battle. Premium time, xp/Commander xp boosters, premium ships, and daily first wins all help gain more Commander xp quickly. Crates, missions, and bureau projects can also give lump sum Commander xp payouts.
Promotion orders are mostly gained by completing the weekly boost and premium supply missions. You can also get them from crates, bureau projects, campaigns, and sometimes sold in the store.