What does "He's full of the Old Scratch, but laws-a-me!" mean in the first chapter of "The Adventures of Tom Sawyer" by Mark Twain?

The phrase characterizes the prevalent Southern dialect during the time in which the story was set.
So, "Old Scratch" is a reference to the Devil; translated, Aunt Polly is basically saying that Tom Sawyer is full of the Devil. Since Tom always gets the better of Aunt Polly, she imagines that he is as mischievous as the Devil.
The other phrase "laws-a-me" just means "Lord-save-me!" Aunt Polly's exclamation demonstrates her frustration as well as her resignation regarding her current predicament. After all, she says that Tom is her dead sister's son, and she can't bring herself to lash him all the time. Sometimes, when she does let Tom off, she feels guilty that she's neglecting his moral training; at other times, when she does lash him, she feels heartbroken for hurting him.
At the same time, Aunt Polly manages to see the humor in her interactions with her naughty nephew. She maintains that Tom knows just how far he can "torment" her before she gets upset, and he also knows that, if he can make his Aunt Polly laugh, a spanking will be less likely. So, Aunt Polly is frustrated but also amused at her quandary; it's no wonder that she exclaims aloud for God's help to raise her lovable, rascally nephew.