Upgrade from 10.5.8 to Mountain Lion
Edit
As pointed out by Kent, an iMac with a 2.16 Ghz Intel 2 Duo seems to be like an iMac (Late 2006) instead of an iMac (Mid 2007).
Technical Specs for iMac (Late 2006)
Technical Specs for iMac (Late 2007)
Therefore, you can only run Lion. Unfortunately, finding a legitimate copy of Lion could be a little tricky if you haven’t already purchased it. If you search for OS X Lion in the Mac App Store you’ll only find Mountain Lion, and Apple no longer sells an OS X Lion USB Thumb Drive (part number MD256Z/A) in its online store.
One of your options is to buy from Amazon or another online retailer.
Or, get Snow Leopard.
Original Post
You have to install Snow Leopard first and then Mountain Lion.
OS X Mountain Lion system requirements
To install Mountain Lion, you need one of these Macs:
**iMac (Mid 2007 or newer)** (Yours)
MacBook (Late 2008 Aluminum, or Early 2009 or newer)
MacBook Pro (Mid/Late 2007 or newer)
MacBook Air (Late 2008 or newer)
Mac mini (Early 2009 or newer)
Mac Pro (Early 2008 or newer)
Xserve (Early 2009)

Your Mac needs:
OS X v10.6.8 or OS X Lion already installed
2 GB or more of memory
8 GB or more of available space

Your model must have a 64-bit EFI boot ROM.
An easy way to tell if you are running a K64 kernel is to use the uname command-line program. The &quot;x86_64&quot; in the excerpt below means that we are running a 64-bit kernel. If the output showed &quot;i386&quot; instead, that would mean a 32-bit kernel.
uname -a Darwin... root:xnu-1456.1.25~1/RELEASE_X86_64 x86_64
Therefore, you need to upgrade to Snow Leopard first.
Other Things
AirDrop and AirPlay will not work on your computer.
Sources
http://support.apple.com/kb/HT5444

Alternate Way
It is possible, but not necessarily legal, to install Mountain Lion over Leopard.
You can look at this.