Hello,

The best source of such data is WorldClim (http://www.worldclim.org/download). There you can find a layers of temperature and precipitation, as well as bioclimatic layers. There are climate variables for current and future conditions (but up to 2070), available at different spatial resolutions (10 min, 5 min, 2.5 min and 30 sec), for different GCMs and for four RCPs. Each layer covers the range of the whole world, so in the appropriate GIS software (eg. GRASS GIS) you have to adjust it to the extent you're interested in.

You can obtain the data from the CMIP5 simulations.

This the document for the data description 

http://cmip-pcmdi.llnl.gov/cmip5/docs/cmip5_data_reference_Appendix1-1.pdf

and you can download the data from Earth System Grid portal 

http://pcmdi9.llnl.gov/esgf-web-fe/

Good luck