This point in the game is the culmination of the relationship between Lee and the other characters, so there's a lot of potential ways it could be altered. The Walking Dead Wikia lists the following for all the characters involved:
The player can choose to go alone on the search for her or ask for help.
- Christa and Omid will offer to help if the player shows the group Lee's bite (or if the player chose to leave an armed Clementine at the house during the trip to Crawford.)
- Kenny will help if the player sides with him in every argument throughout the game and sticks with him through major decisions (alternatively, the player can choose the response to his denial for help, "Clementine is my family." If the player treated Duck and Katjaa in a good fashion before their deaths, Kenny will agree to come along.)
- Ben (if saved) will willingly help if asked to, but will tag along of his own volition if the player had shown him support throughout the past 2 episodes or mention about Clementine supporting him, he will come along.
So it seems that specifically for Kenny, you've got a couple options:
- If you supported him throughout the game, he'll volunteer.
- If you helped Duck and Katjaa through the game, and tell Lee that Clementine is your family, he'll join you.
In my case, I fed and stood up for Duck whenever I could, and I didn't make Kenny pull the trigger when the time came. I was kind and supportive towards Katjaa as well. Kenny didn't immediately want to help me, as I didn't support him in the meat locker and on a few other occasions. However, appealing to his sense of family caused him to come with me.