Are you sure you do need both .rpm and .bin files? Usually providing both would mean they're just "kind" enough to provide a .rpm file that you can install on systems that have rpm (the package manager), or use the .bin installer executable on systems that don't have rpm. The .rpm is preferred if you can use it -- it's easier to remove or upgrade because the package manager can then track the files installed, and also (if dependencies are mentioned) tell you if the package installation was successful or not. Concluded, if you were able to install the rpm (double-click on it or run as root "rpm -Uvh RealPlayer10GOLD.rpm"), the program should be installed - try running it.
The error really is self-explanatory - you're missing the file mentioned. You need to find out which package provides it and then install that package, preferably trough your package manager. Though on RedHat Enterprise Linux you should still stick to the .rpm method..unless of course they tell you to use both .rpm and .bin, which does sound odd
Copying the file to /usr/bin/ directory does not help. /usr/bin is just (usually) included in the users' $PATH environment variable; directories included in that variable (like /usr/bin) are automatically checked for executables if you try to launch something without full path. Therefore the only change you make by copying the file there is that you can run it by typing just the filename (your shell finds it from /usr/bin then) and not with full path, or maybe ./filename if it's in the current workign directory. And of course if it's in /usr/bin it's available to anyone on the system and not just you - probably you don't want to put a setup file there anyway.
See RealPlayer site/documentation/faq/README for information on what software it requires to be installed before installing the player - which package(s) you're missing if you are missing something.
Try if the program runs if you installed the .rpm file. If not, remove the .bin file from /usr/bin as it doesn't belong there and start finding out what you're missing.
I agree there are a lot other (a lot better) players for Linux than RealPlayer; it doesn't have too good reputation if you ask me.