at the Meeting.  

Kit is introduced to Mistress Ashby.  Kit notices that this woman is not dressed as simply as almost everyone else.  In fact, her clothes are quite fine and Kit is impressed.  Then she meets Mistress Ashby's son, William.  He gazes at her and she smiles at him.  After her smile, "William [is] speechless."  He still does not speak as she turns to leave.  After she walks away, he does not move and continues to stare at her.

In a word, Kit is shocked at her uncle's announcement that William Ashby has asked to call on her.  She has actually only seen him once, "for a moment after Meeting", although her uncle insinuatingly remarks that, in her incongruously colorful attire, she had been "conspicuous enough".  Kit "(feels) her cheeks growing hot" as Uncle Matthew and Aunt Rachel discuss the situation as if she isn't even there.  She says nothing as it is apparently decided that permission will be given, even though William's father is a "King's man" in opposition to Uncle Matthew's position as a Patriot.

After Uncle Matthew and Aunt Rachel go to bed, Judith angrily tells Kit that she had been right in predicting that William would come courting.  Kit, remembering, at first feels "a small pleasurable stir of curiosity", but then realizes that Judith had expressed an interest in William as a beau for herself.  In dismay, Kit exclaims that she doesn't want William to come courting her; she doesn't even know him, and she doesn't want to take him away from Judith.  Kit resolves to tell Uncle Matthew of her feelings in the morning, but Judith, admitting that she never actually knew that William had any interest in her, orders Kit not to say anything, and sets her sights instead on John Holbrook (Chapter 6). 

In accordance with the time and the culture, William is courting Kit. His unspoken intentions are not immediately known to Kit. Judith has to explain them to her. When he mentions the work on his house, he is letting Kit know that he will soon be asking her uncle for his blessing in their marriage. He is building a fine house for himself and his future life. All of the time that he spends with Kit shows that she is the one he intends to marry.

William calls on Kit many times, starting in Chapter 7. The visits are always pretty uncomfortable for Kit, because they seem to have nothing to talk about and he seems content to just sit. She doesn't know what is expected of her, and their time together is stiff, literally so in William's case. (Look at how he sits.)
When they do talk, William will sometimes drop deeply meaningful thoughts or observations into the conversation that indicate his future plans, but Kit misses their meaning a lot of the time, and the others, especially Judith, must interpret.

Greg

When William first sees Kit at Meeting, he is impressed and entranced by Kit. Kit is different from any of the girls in Wethersfield, and he is especially impressed by her manner of dress, as she is dressed in beautiful, opulent, and colorful clothes, unlike any of the modestly dressed townsfolk in Wethersfield. Upon seeing her and falling instantly "in love," William asks permission to call on Kit. 

However, when William calls on Saturday evenings Kit finds him boring and has trouble making conversation. William is content to sit quietly and look at her, but Kit is bored and annoyed that he is there most evenings. The only time that Kit shows any admiration for William is when he stands up to Matthew.

William, who is wealthy and of high social standing, takes it for granted that Kit will want to marry him and begins building a large house for them.

When Kit is accused of being a witch, she hopes that William will come to her rescue; however, he doesn't want any association with her that might damage his social status.

Once Kit is acquitted, William returns to call on Kit, but she no longer wants anything to do with him, knowing that she can't depend on him.