What to do to update iMac 7.1 with OS 10.5.8?

You will need to buy a macOS 10.6 Snow Leopard install disc (Apple no longer sells them) and upgrade to that, then update 10.6 to 10.6.8 to get access to the Mac App Store. From there, you will need to download 10.11 El Capitan from here: How to get old versions of macOS - Apple Support

It isn't clear what you're asking.

Are you asking how to upgrade to 10.5.8?

Or are you saying that you currently have 10.5.8 and want to upgrade?

iMac 7,1 dates back to 2007/8, so your support options are going to be limited, but maybe someone can help if we understand what you're trying to do.