Is I don't hardly know correct?

No. Hardly is already negative, so adding don't creates a double negative. Say I hardly know, without don't.

What does hardly ever mean?

Almost never, a frequency expression: I hardly ever watch TV means I almost never watch TV.

Grammar, compared

Hardly means almost not at all, close to zero. Rarely means not often, but still sometimes.

Hardly / Rarely
Hardly	Rarely
I hardly know him. (almost not at all)	I rarely see him. (not often, but it happens)
There's hardly any coffee left. (almost none)	We rarely run out of coffee. (it doesn't happen often)
I can hardly hear you. (barely, almost not)	She rarely calls me. (infrequently, but she does)

Hardly and rarely compared
Point	Hardly	Rarely
Meaning	Almost not at all, barely	Not often, infrequently
Type of word	Degree adverb (how much)	Frequency adverb (how often)
Example	I hardly slept last night.	I rarely sleep less than seven hours.
Common pairs	hardly any, hardly ever, can hardly	rarely, hardly ever (frequency use)

Casi no can become hardly (almost not at all, a degree) or rarely (a frequency), so check whether you mean amount or how often.
Hardly ever is a fixed phrase meaning almost never, a frequency use of hardly: I hardly ever go to the gym.
Hardly already has a negative meaning, so avoid a second negative: I hardly know anyone, not I don't hardly know anyone.
Rarely at the start of a sentence needs inverted word order in formal writing: rarely do I agree with that.

Is I don't hardly know correct?: No. Hardly is already negative, so adding don't creates a double negative. Say I hardly know, without don't.
What does hardly ever mean?: Almost never, a frequency expression: I hardly ever watch TV means I almost never watch TV.

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