I ain’t doing my duty by that boy, and that’s the Lord’s truth, goodness knows. Spare the rod and spile the child, as the Good Book says. I’m a laying up sin and suffering for us both, I know. He’s full of the Old Scratch, but laws-a-me! he’s my own dead sister’s boy, poor thing, and I ain’t got the heart to lash him, somehow. Every time I let him off, my conscience does hurt me so, and every time I hit him my old heart most breaks.
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Got these notes from ChatGPT and Internet:

I ain’t doing my duty by that boy

In modern English, since “do my duty by” sounds a bit old-fashioned or formal, native speakers usually use other expressions with the same meaning, such as:

“I’m not living up to my responsibilities toward that boy.”
“I haven’t done my duty to that boy.”
“I haven’t been the guardian I should’ve been.”
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I’m a laying up sin and suffering…

The “a” doesn’t change the meaning—it’s the same as saying:
“I’m laying up sin and suffering for us both.”
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He’s full of the Old Scratch, but laws-a-me!

It means Tom is as mischievous as the Devil, with "Old Scratch" referring to the Devil. "Laws-a-me" is an exclamation similar to "Lord-save-me!" Aunt Polly expresses her frustration and resignation over Tom's antics, recognizing his ability to both torment and amuse her, making her call for divine help in raising him.
https://www.enotes.com/topics/tom-sawyer/questions/what-does-he-s-full-of-the-old-scratch-but-laws-a-740545

Here are some modern English ways to say “He’s full of the Old Scratch”:

He’s full of the devil.
He’s full of mischief.
He’s a little devil.
He’s up to no good.
He’s a troublemaker.

“laws-a-me” is a folk expression from 19th-century American English, mostly in rural or dialect speech.

Today, people would use modern equivalents like:

“Lord help me!”
“Goodness gracious!”
“Oh my goodness!”
“Heavens!”

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he’s my own dead sister’s boy,

Today people might say:
“my late sister’s son”
“my sister’s boy, who passed away”
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I ain’t got the heart to lash him

It very old-fashioned:

“lash him” for hitting someone is outdated; today we’d say “hit him” or “spank him” (for a child).

Modern phrasing could be:

“I don’t have the heart to punish him.”

“I can’t bring myself to hit him.”
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Every time I let him off, my conscience does hurt me so, and every time I hit him my old heart most breaks.

“my conscience does hurt me so” – using does like this for emphasis isn’t done in modern English. Today you’d say:
“My conscience really hurts me”
“It really hurts my conscience”

“my old heart most breaks” – the most + verb construction for emphasis is also old-fashioned. Today you’d say:
“My heart almost breaks”
“It nearly breaks my heart”
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