You are currently on the latest Safari that your Mac can handle. If you want Safari 6 or later, then you have to upgrade your Mac OS X 10.5.8 to 10.6 or later version of OS X.
To upgrade to 10.6 or later, you have to buy and install 10.6 Snow Leopard first. You can buy 10.6 Snow Leopard discs from here --> http://www.apple.com/shop/product/MC573Z/A/mac-os-x-106-snow-leopard
When you get the discs, just put it on your Mac and follow the onscreen instructions to upgrade.
After upgrading to 10.6.8 Snow Leopard, you can upgrade to Mac OS X 10.10 Yosemite for FREE. You can download and install 10.10 Yosemite from the App Store application in Snow Leopard for FREE.
When you upgrade the OS X the Safari application will also be upgraded.
I'd strongly recommend to upgrade to 10.6 Snow Leopard or later version of OS X not only for Safari but also for some security purposes. 10.5 is no longer safe especially if you are using for internet purposes.
Please note that, this is not my final answer. If you have any questions or would like any additional information, please don't hesitate to ask and I'm happy to continue to help you.