Alter Self allows a character to have Natural Weapons. IIRC, in previous editions, Unarmed Strikes and Natural Weapons were considered Finesse-able (i.e., if you had the Feat, they would use Dex rather than Str). I haven't read anything like that in the PHB (though, unless you're a Monk, it's clear than Unarmed Strikes don't have the Finesse property in 5E), so, was wondering whether the answer is straight up no, or if there's some leeway?
First, the specific case of Alter Self. The Natural Weapons feature of Alter Self explicitly modifies your Unarmed Strike. So if your Unarmed Strike is not a finesse weapon, that doesn't change. This is covered pretty thoroughly in this answer.
However, the general case of 'Do natural weapons have the Finesse property?' is much less clear cut. Take the Raven (Appendix A of the Monster Manual). The Raven has a Str of 2 (modifier -4), a Dex of 14 (modifier +2), and, from the 'Proficiency Bonus By Challenge Rating' table (page 8 of the Monster Manual), its proficiency bonus is +2. Its attack bonus with it's natural weapon (beak) is +4. So it seems clear that it is using
Dex bonus 2 + proficiency bonus 2 = +4 to attack.
There are numerous examples along the same lines, but I chose the raven because the difference between its Str and Dex makes it pretty clear that it's using Dex.
On the other hand, we have the Mastiff. (Also Appendix A of the Monster Manual). It has a Str of 13 (modifier +1), a Dex of 14 (modifier +2), and the same +2 proficiency bonus. Its attack bonus is +3. So it appears to be using
Str bonus 1 + proficiency bonus 2 = +3 to attack.
The Mastiff is not nearly as good an example, but every creature I could find with much higher Dex than Str uses Dex to attack.
Based on these (and a lot of other) examples, it seems clear that some creatures can use Dex to attack with their natural weapons, but some cannot. Alternatively, it's possible that creature attack bonuses are either arbitrary or not calculated the same way as humanoid attack bonuses are, but if this is the case there is no way to answer this question outside of the specific case of Alter Self.
Alter Self's natural weapons feature uses Unarmed Strike. This means that the attacks only get the properties of unarmed attacks. Which means they are a simple melee one handed weapon. They do not have the finesse property.
Alter Self does provide you with the ability to change the damage type depending on how you want the spell to manifest, gives you a much much larger damage die (An increase of 2.5 damage on average), gives an attack and damage bonus and makes them magic.
There isn't anything you can do besides be a Monk to make your unarmed strikes use Dex though.
As far as leeway goes, there's as much as there is with anything else in 5e. You can always talk to your DM about house ruling it. I don't see away around it by the book, so you'll have to make sure this fits your home game's balance profiles. To put it another way, Mearls has been asked a few times about what house rules would break the game. His opinion is basically that Concentration is about the only golden goose in the game.