First, confirm that you did indeed receive a private key. On Bitcoin mainnet, private keys are almost always distributed in a format that starts with either a K or an L (capitalized in either case) and are usually 52 alphanumeric characters long. Here's an example:

L3SD2fM9hzVJyr4zxoHqXHDqHTVfQpbLuWDVm6b6W7PTCSYpkujd

If you instead received a code starting with a 1, a 3, or bc1, then what you received was an address. If you received a hexadecimal code that was 130 characters long and started with 04 or 66 characters long and starting with 02 or 03, then what you received was a public key. None of those are private keys, and there's no way to get the private key from an address or a public key. Here are some examples of addresses and public keys:

1GWLEdJRaMTQSe7NkPAfBCZaR1XdTz4dct
3DXNLK83EiMsseHTfGySGvDL64CKpRTf6G
bc1q4gtqryu6dhs4vzwqwp9fjxegc4rmxcx6shn3qs
03d8f40442a837693820c4ca35bff30036af4df79c5563375200313462086b6d31

If you have any other type of information, that's probably also not a private key and it may have nothing specific to do with Bitcoin.

If you do indeed have a private key, you can import it into most Bitcoin wallets. See the documentation for your specific wallet for instructions (if you can't find the documentation for importing private keys, feel free to ask a separate question).

You don't need any special information to "unlock a private key". In addition, the normal way to receive mining profits is by a normal Bitcoin payment to one of your addresses; I've never before heard of any mining pool paying profits by distributing private keys as that has all the same costs plus a large additional security risk. It seems likely to me that the person you're communicating with is attempting to steal money from you, so be careful in your dealings with them.

If you have the private key you do not need to pay to unlock it.

First you need a wallet and then you need to import the private key.

You can see my previous answer here for some information on choosing a wallet.

Importing the private key will be a different procedure depending on what wallet you choose.

For example, importang a private key in Electrum for an address starting with 3 is here (not the key, the address that the key makes) - it is slightly different if the address where the payment was sent is starting with either a 1 or bc1. If you do not know which address is correct you can import the private key multiple times. You can ask an additional question if you need to, it may have already been answered if you search.

But, the private key may not be a raw private key and may already be formatted for the address type.