In almost every episode, it's shown that they are always next to each other. Also, it is also implied that Jack is convinced that Kim has a crush on him even though she always denies it. He does have a crush on her in return as well. Kim is shown to get jealous when Jack talks about other girls, and vice versa. Jack and Kim have both liked other people (i.e. Donna Tobin, Brad Wolfe, Heather Clark, Brett), and Kim has been known to get jealous when other girls flirt with him.
Sometimes they finish each other's sentences, like in Karate Games. In a promo for the first episode of the show, a voice over stated that they have a crush on each other, and it showed footage of them in a slow motion. It is revealed in Hit The Road Jack that Kim has romantic feelings for Jack. At the end of Wazombie Warriors. it shows the two of them on a date at the movies while watching "The Horse
Tamer", which Jack did not want to see, but Kim did so he went with her to the theatre. They also almost shared a kiss in The Karate Games on top of the Hollywood sign, only to be interuped by the director. In Kickin' It On Our Own they both go on a date together to get pizza.
In the promo for Spyfall the first episode of season 3, it looks like Jack and Kim are a couple. People could assume this because Jack and Kim are seen dancing together. In the episode of Spyfall they have an adorable dancing scene in the end.
They confess their feelings to one another in Two Dates and a Funeral and they officially start dating.
At the end of Wasabi Forever they say their final goodbyes to each other and share a kiss. Afterwards cherry blossoms rain down and Kim gives one to Jack, telling him to keep it as a reminder of her.
Jack and Kim staring dreamily at each other!Source: disneyskickinit.wikia.com