Your best bet is to use the Köppen-Geiger climate classification. It divides the climatic zones worldwide and there is very extensive documentation about its methodology. Should you replicate your study to a different area, you could use this same dataset and give more consistency to your analysis' output. The file is in .kmz but it can be easily converted to .shp in any GIS software or even some websites.
The Köppen-Geiger classification Eric Delgado dos Santos Mafra Lino mentioned is a good choice.
I also found another link where you can download the global climate zones as a shapefile. I am not sure if that is what you are looking for, but have a look at it: