How to request to fly on a flight level, different from the direction, according to the charts?
Say you are flying on a route that requires an even level, but you would like an odd level such as FL350. The phraseology to use would be


  Request FL350


The rest is up to ATC. Depending on local agreements and the current situation, we may be able to grant you the level right away, maybe we need to pick up the phone and talk to downstream sectors or maybe we can't offer the level at all. In any case, there is no harm in asking.

Regarding the part of your question that mentions weather, if you are requesting a level change because of extreme weather (such as moderate/severe turbulence or icing), state so in your request. We will then handle your request with higher priority, and you are more likely to get the different level.


  Request FL350 due to moderate icing