Earlier I wrote more about how to calculate bioclim data layers in GRASS GIS. For people that do not use GRASS GIS, it is probably easier to generate these layers in DIVA GIS. You can get DIVA-GIS for free at http://www.diva-gis.org.
In version 5.2 or higher of DIVA, there is a built in function to calculate the 19 bioclim data layers using the mean minimum and maximum monthly temperatures and the monthly precipitation. There is a tutorial available from CIAT at this link. Note that the bulk of this tutorial is about how to prepare the input data layers; for the steps to generate the bioclim data layers in DIVA, you can go straight to page 4.
The tutorial starts with listing the required software, including ArcGIS. However, ArcGIS is only used for the preparation of the data layers, which could be done equally well with other GIS software such as GRASS GIS or SAGA GIS.