According to the direction, there are correct FL to fly, but due to weather, or traffic, the pilot would request a different FL, from the chart (according to the direction). Is there any correct phraseology to request it?
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
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