Unfortunately, most of the actual descriptions we get of Calpurnia come from the narrator, Scout, and Scout does seem to have a bit of a grudge against Calpurnia because of course the housekeeper expects Scout to behave properly at all times. So, we have to look at what Calpurnia does to determine her examples of kindness, not how Scout describes her.
First of all, we know that Calpurnia is kind because she is connected to Atticus. She grew up and was educated along with Atticus and his brother, and now Atticus trusts her with his most valuable things: his children. Calpurnia is not only a housekeeper but a surrogate mother to Jem and Scout, since their mother died when Scout was quite young. We understand Atticus well enough to know that, despite Scout's beliefs to the contrary, he would not allow anyone that much access to his children if she did not have a kind heart.
Second, we know that Calpurnia is kind to guests--even at the expense of Jem and Scout. On Scout's first day in school, she ends up in a playground fight with Walter Cunningham. Jem invites Walter to come home to lunch with them, and all is well until Walter does something Scout finds appalling. As Walter pours syrup over everything on his plate, Scout makes a rude comment, and that is all Cal needs to remove Scout from the table and give her a stern talking to.
“There’s some folks who don’t eat like us,” she whispered fiercely, “but you ain’t called on to contradict ‘em at the table when they don’t. That boy’s yo’ comp’ny and if he wants to eat up the table cloth you let him, you hear?”
“He ain’t company, Cal, he’s just a Cunningham--”
“Hush your mouth! Don’t matter who they are, anybody sets foot in this house’s yo‘ comp’ny, and don’t you let me catch you remarkin’ on their ways like you was so high and mighty! Yo‘ folks might be better’n the Cunninghams but it don’t count for nothin’ the way you’re disgracin‘ ’em—if you can’t act fit to eat at the table you can just set here and eat in the kitchen!”
Calpurnia does exactly what Atticus needs her to do; she makes every effort to raise Scout and Jem in a way that will make them good citizens and neighbors. In this incident, she does Walter a kindness by removing Scout from his presence; and, whether Scout recognizes it or not, she is doing Scout a kindness, as well. At the same time, when she knows Scout has had a difficult day, Cal makes Scout a special treat.
When Aunt Alexandra speaks dismissively about Calpurnia--in front of her, even--Caplurnia does not respond, though of course she could. That is a kindness toward Aunt Alexandra which works in Cal's favor later because the two women end up working together at the missionary tea. Cal's choice not to respond causes Aunt Alexandra to gain respect for Calpurnia.
After Tom Robinson's trial, Atticus learns that Tom tried to escape and was killed. When he has to go tell Helen Robinson the news, he takes Calpurnia with him not just because she is black but because she is a kind and comforting presence.
In short, Calpurnia is a kind person in all ways. She is strict, but even that is a kindness as she is teaching and reinforcing the lessons Atticus would approve of or is teaching his children himself. She is the perfect fit for the Finch household.