If your Mac shipped with a System Install DVD then boot from it and choose "Restore from Time Machine backup".
If your Mac shipped with Lion already installed you should be able to use OS X Recovery. Boot while holding ⌘ R and choose "Restore from Time Machine backup".
Either one should present a dated list of complete system backups from which to choose. Ideally it should be the most recent one, but if you have reason to believe it is incomplete (it should not be) then try an earlier one.
You can also boot from a locally connected Time Machine backup volume. Make sure it is connected with USB or FireWire and boot while holding an Option key. This will invoke Startup Manager, from which you can select your Time Machine backup volume. It will present the same restore options as OS X Recovery.
These options will be faster and should be less prone to difficulty than using Migration Assistant.
Time Machine and "clone" software use very different philosophies. SuperDuper! is good, so is Carbon Copy Cloner. They are more customizable than Time Machine, whose customizable features essentially consist of an "on / off" switch
Of course that is also its sole advantage over the others - it's boneheadedly simple to use.
Forget Carbonite, as you learned it does not work well on Macs. I know of no similar offsite backup services to recommend.