There are two parts to your question.
First part: how to output time at a fixed location without disrupting other output on the screen.
Low-level approach:
High-level approach: use a text-based UI library, such as curses/ncurses.
Second part: how to update the time display in parallel to other activities.
In the simple case, you can just call time update function periodically from some places in your code you know will be executed regularly enough.
In the more complicated case, you will need to update time from a separate thread of execution. There is a lot said about multithreading, including on this site; unfortunately I can't recommend any specific introductory material offhand, but there are many.
[EDIT] In case you just want to run another program in the background, as @ecatmur suggests, you don't need threads; just use system("program &")
, or fork+exec on Unix-ish systems and _spawn on Windows.