You need the X11 library header files, which are provided in the package separate from the one providing /usr/lib64/libX11.so, with "-devel" at the end of the name. Probably it's called "xorg-x11-devel" or "libX11-devel".

Since you are not root, download this package (manually from a mirror site, or using yum install xorg-x11-devel -y --downloadonly --downloaddir=., if "downloadonly" YUM plugin is installed), extract it somewhere (rpm2cpio file.rpm | cpio -idmv) and append the path to <extracted rpm root>/usr/include to the Makefile.PL parameters, like this:

$ cpan
cpan[1]> look Tk
$ perl Makefile.PL INC="-I/home/username/xorg-x11-devel-extracted/usr/
+include" <other parameters, if needed>
(check if Makefile was successfully generated)
$ make
$ make test
$ make install
$ exit
cpan[2]> exit
$
[download]

If Makefile.PL asks you for some other library which is not installed on the system, you'll have to build it, then install somewhere you have write access to, then add another -I<somewhere> to INC and -L<directory/where/*.so/are/present> to LIBS (read ExtUtils::MakeMaker for more information on this topic).

It's probably easier to ask the administrator to install perl-Tk RPM package.