I have seen a header file cpptcl.h . Is it legitimate? Also where can I find a good quick start example for the same. I am using tcl/tk to design user interface and want to call the scripts by forking in c++ code.
How to connect back-end written in c++ with tcl/tk user interface?
Your question is operating system specific. Because (on many, but not all, OSes) any software application could be made of two (or more) processes cooperating by using inter-process communication facilities, and that approach (which is often implied by using the "back-end" word) has the huge advantage of taking profit from process isolation. Read more about software pipelines, about pipes and sockets. They might be available on your target OS. For a good overview on operating systems, I strongly recommend reading Operating Systems: Three Easy Pieces (it is freely downloadable).
If you are coding for Windows, study and learn the WinAPI. If you are coding for Linux, study its programming API by reading ALP and looking into syscalls(2) and every system call referenced from that man page. Both documentation are useful to code inter-process communications.
You could write a single-executable application, by linking a Tcl/Tk library (or code) with your core, application specific, C++ code, but if you do that, you won't speak of any back-end (or front-end), but of the application-specific layer and GUI user-interface (abstraction) layer of your code.
At last, Tcl/Tk seems to be out of fashion for GUI development. Today, a good C++ GUI toolkit is Qt, which is vastly superior (in features and in quality of the look and feel of the interface) to Tcl/Tk.
I have seen a library cpptcl.h.
That is not a library, but a header file (for the C & C++ preprocessor) declaring the interface of some library. I guess that the corresponding library would be a dynamically-linked library, e.g. some lib*.so
ELF shared object on Linux, or some *.dll DLL on Windows.
Is it legit
I suppose you mean "is it legitimate"? Surely yes. Or "conforming to rules" (then of course yes). You could ask about the legal status of that library, and the right wording for such a question is asking about its software license. Tcl/Tk are open source software with some BSD license.
where can I find a good quick start example for the same.
Both Tk and Qt have excellent documentation (that you should read during several days ...). And the documentation of Tk is impressive (and so is the documentation of Qt). I recommend reading the Tk tutorial first (and if you choose Qt, read first its Getting Started with Qt documentation). Most GUI software involve some event loop (which is provided by Qt and probably also by Tk). You probably should have callbacks (and Qt uses signals and slots) from that event loop to C++ functions of your applicative layer.