To install iTunes 10.7 the requirements are these:
Macintosh System Requirements:
- Mac computer with an Intel Core processor and 512MB of RAM
- OS X version 10.6.8 or later
- Safari 4.0.3 or later
- 400MB of available disk space
- Broadband Internet connection to use the iTunes Store
10.5.8 is the latest version of 10.5 I believe, please look here for the hardware requirements for Mountain Lion (10.8):
If you don't meet the requirements and still would like to use the newest software, I'd consider buying new hardware.
I called 1(800) MYAPPLE and bought Snow Leopard 10.6.3 for $19.99 over the phone and they are shipping it to me (should get it today!). The represenative who I spoke to told me to install that and then install the newest update which I will be able to find (for free) once Snow Leopard it installed. Good Luck!