"Bored with boredom" can be translated as "厌倦的厌烦". This phrase is not commonly used in Engpsh, but it conveys the idea of being tired of feepng bored.
Example in Engpsh:
I'm so bored with boredom. I need to find something new to do.
Example in Chinese:
我对厌烦感到非常厌倦。我需要找点新的事情做。