You need to have crew training tomes to upgrade your crew. I'm guessing you used up the ones you had. The crew page will tell you how many crew training points you have available. Training points for crew can be had as quest rewards, or purchased from training tome venders in the taverns, or in a few rare places you might find one lying around as you are questing. They look like open books. Keep in mind that basic tomes when you are low level are cheap, but as you advance, they get more expensive. Also the closer your crew gets to your level, the less the training tomes train them. I like to let my crew lag enough behind my level that a training point will level them up at least one level sometimes 2. Your first mate is a good one to put more training into, because they are with you in every fight.
Some crew, like Ratbeard, have promotion quests; when you raise them to a level where they are eligible to begin a quest, they will tell you that they have some news, or have heard from an old friend, or whatever, and ask to talk to you in the basement of a tavern. Any tavern will do. Take the crewman there and he or she will talk to you and a new quest will be added to your list of open quests (and it will show an image of the promoted crewmember in the "rewards" part of the quest box for that quest if you check). Promotion quests are pretty much limited to the crew that talk to you (who are also pretty consistently the strongest crew).
Other companions can be promoted for gold. When you train them to a level where they are eligible for promotion, you will receive a message that one of your crew is eligible for promotion. Again, go to a tavern basement and talk to that crewman - you will be told how much gold it will take to promote the crewmember.
Either way, after the promotion remember to go to crew management (U) and give the crewmember the new epic feat(s) he or she has earned with the promotion.
If any crewmember is already at a level where they can be promoted, they will have an exclamation mark on their icon/portrait in the crew management roster.
The guide that Sniper Rose & Chrissys Blessings posted is really good to let you know at what level your companion will promote (gain epics) if at all.
There is a difference between training & promoting your crew. At certain levels ALL of your companions will reach a "training" stage where you can pick additional Talent stats (Accuracy, Dodge, Rough, Armored, Agility, Will, Strength, Health, etc) for them.
At "promotion" stages you will either have a quest to send your companion on or else have to pay gold to promote them. You will be notified that it is time to promote a companion because they will want to talk to you in a Tavern. After you have completed their quest or paid the gold then you will be able to choose an Epic (Relentless, Burstfire, Mojo Rising, Riposte, Double Tap, etc) for them.