How do I obtain more information on my deceased next-of-kin's bank accounts?

You would have to contact the bank(s) as a next-of-kin and explain that your loved one has passed on and you would like to check if they held an account with the bank. Do note that each bank may have its own processes with regard to disclosure of this information.
Unclaimed monies in bank accounts will continue to be held by the banks even when the accounts are inactive or if the account holder has passed on. As the banks are required to comply with customer information confidentiality requirements under the Banking Act, the banks are unable to disclose details pertaining to the bank account(s) held by the deceased without the appropriate court documents such as the Grant of Probate (if the deceased made a will) or Grant of Letters of Administration (if the deceased did not make a will).
You may check the Family Justice Courts website or enquire with the Family Justice Courts for more information on applying for these court documents.