Let's assume that we know the language of the plaintext. If you use the Friedman test on this Playfair ciphertext, the ciphertext will show a low index of coincidence because the same plaintext letter can be encrypted to various values.
You could use the Friedman test as one indicator to you whether you are dealing with a Playfair cipher (The others being: an even number of letters; perhaps the "J" is missing; and no digraph is made of the same letter, e.g., "EE", after you break the ciphertext into digraphs.)
Don't worry about the absence of the letter "J" because that is normal in the traditional Playfair cipher.
For cryptanalysis of Playfair, see this SE Crypto answer and this lucid explanation of how to get at a Playfair keyword.