Maybe I'm missing something, but I've reread this dark "gift" multiple times and I don't see the benefit to it. Since dark gifts can be a choice, why would anyone choose this? If it's supposed to be like Jeckle and Hyde then shouldn't the be some benefit to the new form like extra strength, or a bite or claw attack? What am I missing?

The alter self spell cuts off in the van richtens link, i had wondered the same thing, look at the actual spell description

 Alter Self

Sounds more like a Role Playing reason to take it rather than cool damage bonus reason.

Yes, I understand it let's you use one option from alter self, it just seems pretty useless, since it's always the same monstrous form but gives to no mechanical benefit, unlike the other dark gifts.

all you get its claws bite or movement

Quote from WulfRanger >>

all you get its claws bite or movement

 You don't get claws or a bite. Alter Self gives the caster three options when casting, and only the Natural Weapons option gives you an unarmed attack like claws or a bite.

Second Skin specifies that you only get the Change Appearance option from Alter Self, and you only change in to the same form each time you change. Even if aesthetically that form looks like it has claws or fangs, they give you no mechanical bonus.

From Second Skin: "Transformation. You can cast the alter self spell to appear in your second form. When you do so, you gain the effects of that spell’s Change Appearance option..."

I think all of the Dark Gifts are a bit flawed. In the same way there is no real mechanical upside for the Second Skin, there is no mechanical downside to Touch of Death. Second Skin seems designed to be one of the options that a DM can give the player as a forced option. One of those 'You make this bargain thinking you'll get great power but JK you get nothing of substance' sort of things. 

That said, I do think it is interesting as a tool for RP, but there are plenty of players and DM that won't consider it because why do you need to have a weird mechanical caveat to do some RP thing.

Many DMs are new, inexperienced, or often have to be ruthless with their players because the edition encourages players to go beyond the rules - often a little too much.  A Dark Gift like Second Skin gives the DM a mechanical thing they can offer a player that explains the limits of the ability, and also gives DMs ideas for what their own homebrew Dark Gifts might be.  The whole list is primarily examples of what these generally look like, with the obvious expectation that DMs who're more comfortable with stepping outside the rules will create their own.