I use dumbbells for almost all of my bicep training because they afford several advantages. Dumbbells allow a greater range of motion, whereas a barbell or EZ-curl bar locks you into a more rigid arc of execution. Dumbbells also allow you to supinate your wrists to hit your biceps from a variety of angles.
Another advantage is that dumbbells allow you to concentrate fully on your biceps and get a complete workout without overtaxing the rest of your body. However, barbells allow to lift maximum poundages so I would never recommend that you discard them altogether. None the less, they bias your body's center of gravity forward, causing you to expand energy stabilizing yourself. In that regard, then, barbells are not what you would call efficient.
I begin with standing alternate curls because they take more energy than say other bicep exercises. Of all the dumbbell movements I will talk about they are probably the best mass builder. I like to start off with a light warm-up set. I cannot stress enough the importance of warming up, before you get into your all out working sets. I will then bump up my weight and do 3 working sets. Things to remember when doing tis exercise is to concentrate on a steady smooth movement. Also when brining the weight to the top of the movement, try turning your little finger in toward your body a bit. This will provide you with supination which will allow you to fully peak your biceps.
Next I move on to single arm preacher curls, using a moderate weight. You want to have enough weights that you can hit about 10 reps per arm per set. Make it a point not to let the dumbbell twist throughout the movement, keep it parallel to the ground. Also concentrate on slowly lowering the dumbbell back down once you reach the top. This exercise if done correctly will really hit the belly of the exercise.
Lastly I will perform seated concentration curls. This is a good mass and blood-pumping movement to finish with. The focus when doing these curls is directly on the biceps, whereas you have put a lot of effort into stabilizing the weight with the standing version. In this you want to try and hit the brachialis, the muscle between the biceps and the triceps.
Start the movement with your arm extended and bring the weight up across your body, almost toward your opposite shoulder. This is very different from the standing concentration curls. Also, keep your palm turned forward your body throughout the movement, raising it in a hammer-like motion that will target your outer bicep.