Use different rep ranges. For instance, if you're doing barbell curls, load on the weight and shoot for 6 reps or so, to positive failure. I'm talking about 6 quality reps of the most weight you can use while keeping the best possible form. Start with the barbell (or curl bar) fully extended in front of you, then, in a wide arch, curl the weight up to shoulder level, but don't throw it back towards your shoulders to rest; instead, squeeze the biceps at the top for a second or so. This prevents the muscles from resting, which takes tension off the muscles, which is something you want to avoid when doing curls. Lower the weight back to the starting position through the same wide arch, slowly (2-3 seconds). Pick a weight where you can only do around 6 reps. Every one in a while I'll do a couple reps of cheat curls at the end of a set of heavy barbell curls; when I can't get any more strict reps, I'll use just enough momentum to curl the bar upwards, still contracting the biceps at the top, and still lowering the weight 2-3 seconds. Though your biceps won't be strong enough to curl the weight in a strict manor, you're still making them work as hard as possible. I don't do this every time though. Anymore I usually only do a couple sets of each type of biceps exercise.
Now pick another biceps exercise. I recommend choosing from the following: incline DB curls, cable curls, preacher curls, alternating DB curls, ect.. Do a couple sets of these, but do them in a higher rep range, such as 8-10. Still, focus on complete range of motion, squeezing the biceps at the top, and lowering slowly (2-3 seconds). I prefer incline DB curls, starting with the dumbbells hanging by my sides, palms facing each other, and supinating (turning the wrist to where the palms face up, as if you're holding out a cup of soup) the wrist as I curl the dumbbells upward. At the top of the movement, supinate your wrists so that you're trying to get your pinky fingers higher than your thumbs (you may not actually be able to get your pinky fingers higher than your thumbs, but the point is to supinate the wrist as much as possible; the function of the biceps are to curl the arm and to aid in supinating the wrists). A couple sets of this or another biceps exercise at 8-10, full, quality reps, should be sufficient.
If you want, finish with a set of chins with slow negatives. I usually just do one set of chins. Or you could even do chins right after the heavy barbell curls and finish with a set of your choice of biceps exercises. I don't recommend any more than 5-6 TOTAL sets for biceps. The biceps are a small muscle that get a lot of work when you're doing back movements, so they don't need a lot of specific work for overload. I do think some biceps specific work is necessary, but not a lot. Trail and error my friend and find your sweet spot, but trust me, too little is better than too much.
So the keys to biceps specific training are full range of motion, curling the weight through a wide arch, contracting the biceps at the top of the lift and lowering the weight slowly to the starting point. That pretty much holds true for any biceps exercise. Another tip: don't concentrate on how much weight your lifting; instead, focus on what the muscles are doing. The weights are a means to the end and your focus should be training the muscle, not just lifting the weight. A lot of people only focus on lifting the weight, period, no matter what the form and they're usually the same people with unimpressive physiques. If you're still not a little sore the next day, don't worry about it too much; soreness is not a good indicator of muscle overload just by itself. I guarantee if your diet is good and you're training methods are sound, you will see results, even if you're not super sore all the time.
Also, I like flexing the biceps right after a set. This pumps more blood into the muscle, which provides added benefits: prevents the muscles from losing the pump prematurely, therefore I can continue to escalate the intensity with each subsequent set. Also, after your completely done with your biceps specific work, stretch them out.
Happy training!