Oh my good!
Ok, first of all I would strongly suggest any other form of transfer than serial. But it depends on how much You want to transfer, in which direction or whether learning about serial communication is the main objective of Your undertaking, rather than just copying some bytes.
And we need to know more. Does Your DOS-machine have floppy, CD, USB, (I assume it doesn't have a network card?), is it very old
, does it have any Windows 3.* on top of it, does it have to keep running DOS5, will You use it as a (serial) terminal to the Linux-machine, have You as of now had any working contact over serial... and so on.
Because this will let us know more about what options are available.
Best regards,
Lakris