Mrs. Wilkins (and the "Mrs." here just means that she's older, not that she's married) is Squire Allworthy's servant. She's the one who first takes care of baby Tom when Squire Allworthy finds him wrapped up in his bed. But don't go thinking, just because she helps to look after young Tom, that Mrs. Wilkins is some kind of warm and fuzzy type.
She doesn't have a huge role in the novel, but most of her purpose in the story seems to be to show how ungrateful and snobby servants can be.
When Squire Allworthy believes he's dying, he tells Mrs. Wilkins that he plans to leave the servants some "tokens to remember [him] by" (5.7.17). She bursts into tears, but she definitely isn't crying because it appears that her employer is about to die.
No, what bothers Mrs. Wilkins is that Squire doesn't distinguish between Mrs. Wilkins and the other servants. She swears, "now we are all put in a lump together. If so be that it be so, the legacy may go to the devil with him that gave it" (5.8.1). In other words, if she's going to get the same amount as the other servants, then damn that inheritance anyway (though she still plans to keep any and all money she can get from Squire Allworthy, living or dead).
Mrs. Wilkins resents Squire Allworthy's efforts to be generous. When he tries to be nice, she immediately assumes that he could have been nicer. The novel suggests that it's human nature for servants to resent their employers, so it makes sense that Mrs. Wilkins despises Squire Allworthy in secret ("The devil shall wait upon such a gentleman for me" (5.8.1), she exclaims).
But what is weird about Mrs. Wilkins's understandable class resentment of Squire Allworthy is that she, in turn, makes sure to maintain her position above other, lower-ranked commoners. Not only does Mrs. Wilkins hate that Squire Allworthy hasn't given her more money, but she also gets disgusted that all the servants have been "put in a lump together." She wants to be singled out as the best, the highest, of the working class. If she can't be a boss, she at least wants to be at the top of the hierarchy of employees. Mrs. Wilkins has no real power over Squire Allworthy, but she makes sure to use her place in his household to boss around the other servants and villagers below her.
We see this bullying side of Mrs. Wilkins in her treatment of other women, in particular. It's Mrs. Wilkins who decides that Jenny Jones must be Tom's secret mother. When Mrs. Wilkins first "asks" Jenny if she's Tom's mom, she addresses her outright as an "audacious strumpet" (1.6.12) (which means "cheeky prostitute"). Clearly, she isn't exactly willing to give Jenny the benefit of the doubt.