Transferring a cassette to Logic is no problem - just treat it like any other external analogue audio source you may wish to record. Connect the outputs of the cassette player to your mixer or audio device, select the appropriate inputs in logic, set levels and record.
Fixing the pitch of out of tune vocals is a different matter. I presume you are referring to a mixdown of music with vocals, requiring seperate treatment of the vocal without changing the pitch of the rest of the music? If so, Melodyne editor would be the tool of choice. This can run as an Audio Unit plugin in Logic.
After recording your songs from the cassette into logic, you would need to transfer the audio into the melodyne editor and start tweaking. It's a very advanced algorythm, and requires some practice to get the best out of.
As Mark says, you'll need Melodyne - but there are no guarantees that you'll be able to isolate the vocal completely. If there are any instruments that play in the same frequency range as the vocal, you may not get a clean result.
Yes, Colin is quite right to mention this. Obvious examples of what Melodyne editor could not be expected to achieve would be retuning one violin in an orchestra, or a single out of tune voice in a choir. It would make sense to try the demo first.