What is the correct phraseology to request a climb?
I would say that “Request climb FL350” is better.

I never use the word “to” in altitude requests as it can be confused with “”two” as in “request climb to seven thousand” (request climb two seven thousand) 

If you get in the habit of using “to” with flight levels, you will probably use it in a baro altitude request as well.

I also leave out “climb” and just say “request FL350”. But, when ATC says “climb FL350”, I will read back “climb FL350”.