The relationship between the 2 phenomena is explained well here:
One of the most common mistakes that we have observed in discussions of climate and atmospheric change is confusion between the rather separate concepts of ozone depletion and global warming. This isn't necessarily surprising given the scant information that most people pick up from the media. However, for many years meteorologists have been fighting a rearguard action to persuade people that the globe isn't warming because there is more sun coming through the ozone hole. There are however important connections between the two issues that complicate potential actions that we might take to alleviate the different problems. This week, for instance, a new IPCC report was released that looked at the greenhouse warming potential of many of the replacement chemicals (HFCs and HCFCs) that were used to replace CFCs in aerosol cans and refrigeration units under the Montreal Protocol (and subsequent amendments).
The connections actually go both ways: Firstly, CFCs, HFCs etc. are greenhouse gases, while ozone is both a greenhouse gas and an absorber of solar radiation (in the UV range), and so changes to their concentrations affect the radiation transfer through the atmosphere. Secondly, the chemistry that controls ozone loss is very sensitive to the local temperature and humidity, and as that is affected by climate change, that will impact ozone depletion as well.
The original CFCs were powerful greenhouse gases (about 0.34 W/m2 forcing since 1850), and even allowing for a cooling due to the subsequent depletion in stratospheric ozone (-0.15 W/m2), they had a net warming effect. Therefore the ongoing phase-out will help both the stratospheric ozone problem and reduce the forcings leading to global warming. CFC concentrations are indeed now starting to level out and are expected to decrease further in coming decades. However, some of the replacement gases (for instance HFC-23) which are not as harmful to ozone, nonetheless have an significant greenhouse warming potential. The total forcing from these replacements is expected to be small compared to increases in CO2, but any reductions that can be easily made can potentially offset some increases in CO2. Thankfully, some other replacements exist (for instance ammonia) which neither affect ozone nor the greenhouse effect. The cure for ozone depletion has not turned out to be worse than the disease!
On the other hand, some of the climate change effects on ozone were discussed previously in connection with Arctic ozone levels. These effects are both chemical and dynamical. The chemical impacts relate mainly to increasing levels of methane and stratospheric water vapour directly affecting the local chemistry. Additionally, stratospheric cooling (caused by increasing CO2 as well) has an indirect effect on the rates of many of the ozone-destroying reactions (accelerating ozone loss). Dynamically, planetary and gravity wave activity, (related to convection and the jet streams, for instance) all affect the momentum balance in the stratosphere and control the Brewer-Dobson overturning circulation. Therefore changes to those can potentially affect the stratospheric circulation and thus change stratospheric winds and stability. These dynamic effects can often lead to local changes of temperature (particularly in high latitudes) much greater than any radiative change e.g through possibly changes to the strength of the polar vortex (Shindell et al, 1998).
http://www.realclimate.org/index.php?p=145