How to run earth system, land surface and climate models?
As alluded to in some of the comments, all of the CMIP5 models will have been run on supercomputers, and it takes a lot of effort to get one of them running on a new platform, even for a team who know both the model and platform well already.  In addition to that, many of those models are closed-source and are not generally available to an individual researcher.  However, there are several older models such as MITgcm and EdGCM that it might well be possible to get running on a cluster.  It's something I've thought about but never quite got around to.  One group have even been
running CESM on Amazon's EC2 service.

Depending on what aspect of leaf phenology you're interested in, you may need to run quite long simulations (e.g., several centuries), which is prohibitive with a full ESM.  You can make a lot of progress in this area by running just the land surface components.  This is several orders of magnitude simpler than running an ESM: the source codes are often freely available where their host ESM code is not, and they can be run globally on desktop machines or clusters.  Some example land surface models you might want to look into:


Community Land Model (CLM) (used in CCSM, US)
JULES (used in HadGEM, UK)
CABLE (used in ACCESS, Aus)
ORCHIDEE (used in IPSL-CM5, Fr)
SURFEX (used in CNRM-CM5, Fr)


Each of these will require a set of forcing boundary conditions, and there are several commonly used by the land community:


GSWP3
WFDEI
CRU-NCEP
Princeton


If you go down the land surface route (and I suggest that's a good place to start), you should probably just pick a model and ask it's community directly about the details of getting it running.  In my experience, several of them have very good tutorials especially for people with a bit of Linux experience.