Is it grammatically correct to combine two negative ideas into one sentence and then start the second part with “neither”, like, "I don’t usually wake up at 6 a.m., neither do I like to wake up at 5 a.m.”?
It is grammatical. But it is preferable to remove the comma splice: replace the comma with a semicolon or a period.
In my view it is grammatical, but rather literary. Some people would object to combining these into a single sentence, but I find it OK.