No, I'm sorry you can't. Your iPod touch simply does not have the hardware to run later iOS versions. You can't make apps that require a higher iOS version work on older firmwares either, as they are closed source.
Taken from the Whited00r 6 - FAQ & Troubleshooting - READ BEFORE POSTING!
Why can't I install Apps that Requires iOS 4.3 or upgrade above iOS 4.2.1?
iOS 4.3 and above and apps that requires iOS 4.3 are using binaries CPU instructions sets for newer devices equipped with ARMv7 CPU & faster GPU and only those Apple devices have it:
ARMv7s = iPhone 5, iPad 4
ARMv7 = iPhone 3GS, iPhone 4, iPhone 4S, iPod Touch 3G, 4G & 5G, iPad, iPad 2, iPad 3 & iPad Mini.
Apple Devices with the ARMv6 CPU supported by Whited00r & App Time Machine but not by Apple anymore and dropped slowly & surely by iTunes
iPod Touch 1G or 1st Generation - max iOS 3.1.3
Original iPhone 2G or 1st Gen - max iOS 3.1.3
iPod Touch 2G or 2nd Gen - max iOS 4.2.1
iPhone 3G or 2nd Gen - max iOS 4.2.1 but it's dog-slow, the very limit of what that hardware can take. Whited00r is based on a tweaked iOS 3.1.3 because it is much more fluid with plenty of free available RAM to run apps.
Apple could easily require that app developers include iOS backward compatibility to be eligible to iTunes App Store but they decided not to. Apple makes the Xcode software that developers are using to create apps and removed support for ARMv6 binaries compatibility in their Xcode 4.5 version (see Mac Developer Library - Xcode Release Notes for technical details). Apparently they re-enabled ARMv6 binaries compatibility in the current Xcode 4.5.2 according to this.
It's just like computers, you don't except older computers to be able to run the latest games, even if it doesn't have large requirements hardware wise, it could depend on software that can only run on newer hardware (complex explanation for that but CPU and some other things can be the reasoning). Computers have evolved at a slower pace than mobile phones, in terms of what can be done/dropped support. It's just how things work in the mobile world.