[He's dedicated, but not to breaking the barrier. He loves Kitty in whatever weird, nebulous way he tends to love the people he gets close to. He wants good things and happier times for her, and maybe some of that can be conveyed in a kiss.
That she happens to be clever and brave and lovely muddies the waters somewhat. There's certainly attraction in addition to affection.
Chekov is quickly forgetting what he was worried about because everything is Kitty and getting as close to her as possible is of the utmost necessity. It's more important than breathing.]