I was practicing Hangestu this morning with a lot of focus and tension. Throughout the final move I kept the tension, with my muscles working against each other, and felt a definite pain within my biceps. I was wondering if anybody else had encountered this, and if it is possible to injure yourself by doing this. Is it beneficial to do Hangetsu with a lot of tension? It's not a kata I have trained much in the past.
If Hangetsu is really similar to the Seisan I know, it is indeed a kata traditionally performed with a lot of tension. Most of the tension should be in your core however, as if you were preparing to get punched and wanted to reduce the damage. While your arms should be tensed, they shouldn't be as tense as your abdominal muscles. One of the goal of seisan is to teach you to put your core into your blocks and punches. Combined with a good leg stance and proper weight shifting, you'll be able to generate a lot more power compared to simply using your arms to perform these movements.
What you describe in your question looks a lot like improper stretching to me, though. If you are performing the kata with heavy tension in your arms and legs, you're putting your muscles under constant strain. That kata is also done pretty slowly, too, with short bursts of speed and power where it's easy to put even more strain on your already tense muscles.
You seem to imply doing katas as part of your morning workout. Do you stretch beforehand? If so, do you give enough attention to your biceps? If they aren't warmed up properly, it's entirely possible that the pain you're feeling is due to the constant strain you're putting your unprepared muscles under. Ask your sensei (and maybe a certified trainer if you have a gym membership somewhere) for proper bicep stretching techniques and try to perform this kata while properly warmed up.
If the pain persists, try to talk your sensei into looking at your stance, posture, movements and technique carefully. You mention not having much experience with this kata, and this could be why you're feeling some pain, as it's not impossible that you're simply doing the movements slightly off. Correcting your technique might alleviate the strain on your muscles. I'd still suggest investigating proper stretching techniques, though. Usually can't go wrong with that anyways.