So my wife bought me the Destiny box for Christmas (the $29 dollar one that has Kylo and Rey in it) and we looked at the starter video and played a couple of quick rounds to get the hang of things. My question is about the die include in this box. When I got to the "Customization" section of the video it mentioned when customizing a deck you can choose 2 die for certain characters. This boxed set only had one die each for the 4 characters included. Is there a way (or need) to get the extra die for these characters? Or will they always be a one die card?

Buying a second set is the way to get a second dice for the characters.

For competitive play, generally speaking people will try to play a character with two dice (Elite). Not always, but it's common. Whether it's needed for how you want to play is up to you.

I bit the bullet and bought 2 starter sets.

That also gives you enough cards for 2 tournament legal 30 card decks.

Don't forget the points change on Captain Phasma that happened in the latest FAQ that makes her (elite) and Kylo (elite) over the 30 point limit for tournament use.

Edit to add that if you are only going to play at home with the family it doesn't matter.