1. KNOW THE DIFFERENCE BETWEEN A PULL-UP AND A CHIN-UP. A pull-up grip is with palms facing away (pronated), outside shoulder width. A chin-up grip has the palms facing you (supinated) with hands 8-12" apart. What these positions have in common is that when your chin clears the bar, the forearms are perpendicular to the ground. They also both involve a strong contraction of the lats, the muscles of the rotator cuff, the biceps and the forearms. Where they differ is that the pull-ups uses shoulder adduction (pulling elbows down and back from the sides), and chin-ups use shoulder extension, in which elbows come down and back from the front. You should train both equally, but most people will find it easier to get a chin-up first, as the bicep can flex the elbow a bit more efficiently from that position.
2. LOSE THE NATURAL WEIGHT VEST. It's entirely possible that right now, this minute, you're strong enough to do an unassisted strict pull-up, and it's only the extra 10 or 15lbs you're carrying that's foiling you. Just like 5lbs too much on the bar can mean the difference between hitting a PR and missing a lift, excess body weight can be the difference between success and failure on the pull-up bar.
Get serious about adopting a coherent nutritional strategy, and then follow it until you are at your optimal weight. Having a performance goal can transform deprivation into sacrifice. Deprivation is when you are denied something; sacrifice is when you give something up to get something better.
3. STOP KIPPING ON THE BAND. Just stop. Kipping is for after you get your first unassisted strict pull-up. Only then are your shoulders strong enough to remain stable under the enormous force kipping imposes on them. Kipping on the band won't get you closer to your goal, and it's probably not conducive to your long-term shoulder health (although there's no definitive proof about that). So just...stop it. Now. Yesterday. DO IT.
From now on, until you have strict unassisted pull-ups, ALL your pull-ups should be performed strictly. Alter the assigned volume if necessary, as well. How many pull-ups, then, with what band? Lightest band possible, no more than two sets per round. If you get all the pull-ups, awesome. Next time, use less assistance. If not, get as many as you can in two sets and drive on. Next time pull-ups come up in a WOD, stay at the same assistance and do the work in no more than two sets per round.
4. GREASE THE GROOVE. Put a pull-up bar up in the door to your kitchen. Every time you walk under it, try for a pull-up. Strength is a skill as much as a physical adaption. Constant practice is the key. More about this here: http://humanmachine.wordpress.com/.../grease-the-groove.../
5. GO NEGATIVE TO GET A POSITIVE RESULT. The use of rep negatives (timed eccentric contraction from top position to bottom) is a great way to build the additional strength you'll need to do a pull-up. Try this:
Every other minute on the minute for ten minutes: with a pull-up or chin-up grip, do a jumping pull-up to get in finish position: chin cleanly over bar, head held neutral, midline hollow and stable, legs straight, feet together. Hold for five seconds. Rest for fifteen. Then jump back where your elbows are bent at 90 degrees. Hold for five seconds. Rest for fifteen. Jump up to finish position, and then take a full 5 seconds to lower yourself to bottom position (remember to keep shoulders active!). Hold for five seconds. Rest for the remainder of the minute, plus the next minute, before going again (total of five sets).
Talk to one of your instructors before implementing this accessory work, and don't do it more than once a week. Don't do it on a day that has pull-ups in the WOD, or on a day in which we're going to row. Negatives are powerful medicine, and can leave you very, very sore!
Once you can complete the five sets without going to failure, add weight by holding a dumbbell between your feet or using a dip belt. With consistent practice (and strong attention to your nutrition), you should have your first unassisted strict pull-up within two months. Good luck!