How can I convert a SHA256 hash into a Bitcoin base58 private key?
What you are requesting is described as computing the Wallet Import Format for that private key:


http://en.bitcoin.it/wiki/Wallet_import_format


Using your example:

1.) Take a private key (Below is the HEX representation of binary value)


  7542FB6685F9FD8F37D56FAF62F0BB4563684A51539E4B26F0840DB361E0027C


2.) Add a 0x80 byte in front of it


  807542FB6685F9FD8F37D56FAF62F0BB4563684A51539E4B26F0840DB361E0027C


3.) Perform SHA-256 hash on the extended key

$ echo -n '807542FB6685F9FD8F37D56FAF62F0BB4563684A51539E4B26F0840DB361E0027C' | xxd -r -p | sha256sum -b



  7DE4708EB23AB611371BB778FC0C8BDE80394AB2D8704D7129FB5771E2F1730D


4.) Perform SHA-256 hash on result of SHA-256 hash

$ echo -n '7DE4708EB23AB611371BB778FC0C8BDE80394AB2D8704D7129FB5771E2F1730D' | xxd -r -p | sha256sum -b



  CD5C4A8E03DFBB0E3AA021C2D74A9EAA43CE4C9CB1B20FC88729A7A5834141CA


5.) Take the first 4 bytes of the second SHA-256 hash, this is the checksum


  CD5C4A8E


6.) Add the 4 checksum bytes from point 5 at the end of the extended key from point 2


  807542FB6685F9FD8F37D56FAF62F0BB4563684A51539E4B26F0840DB361E0027CCD5C4A8E


7.) Convert the result from a byte string into Base58 to get it into the Base58Check format. This is also known as the Wallet Import Format

(Converted from point 6 to base58 using https://bitcointools.appspot.com, or using encodeBase58 from https://github.com/grondilu/bitcoin-bash-tools)


  5JhvsapkHeHjy2FiUQYwXh1d74evuMd3rGcKGnifCdFR5G8e6nH