There are a few connections between the two, but they are largely separate issues.
First, it's important to know that ozone plays two different roles in the atmosphere. At ground level, "bad ozone" is a pollutant caused by human activities; it's a major component of health-damaging smog. The same chemical occurs naturally in the stratosphere, and this "good ozone" acts as a shield, filtering out most of the ultraviolet light from the Sun that could otherwise prove deadly to people, animals, and plants.
The ozone hole refers to the seasonal depletion of the ozone shield in the lower stratosphere above Antarctica. It occurs as sunlight returns each spring, triggering reactions that involve chlorofluorocarbons (CFCs) and related molecules produced by industrial processes. These reactions consume huge amounts of ozone over a few weeks' time. Later in the season, the ozone-depleted air mixes with surrounding air and the ozone layer over Antarctica recovers until the next spring. Other parts of the globe have experienced much smaller losses in stratospheric ozone.
Because of international agreements to limit CFCs and related emissions instituted with the Montreal Protocol, it's expected that the ozone hole will be slowly healing over the next few decades.
The ozone hole does not directly affect air temperatures in the troposphere, the layer of the atmosphere closest to the surface, although changes in circulation over Antarctica related to the ozone hole appear to be changing surface temperature patterns over that continent. Ozone is actually a greenhouse gas, and so are CFCs, meaning that their presence in the troposphere contributes slightly to the heightened greenhouse effect. The main greenhouse gas responsible for present-day and anticipated global warming, however, is carbon dioxide produced by burning of fossil fuels for electricity, heating, and transportation.
Higher up, the loss of stratospheric ozone has led to some cooling in that layer of the atmosphere. An even larger effect comes from carbon dioxide, which acts as a cooling agent in the stratosphere even though it warms the atmosphere closer to ground level. This paradox occurs because the atmosphere thins with height, changing the way carbon dioxide molecules absorb and release heat. Together, the increase in carbon dioxide and the loss of ozone have led to record-low temperatures recently in the stratosphere and still higher up in the thermosphere. Far from being a good thing, this cooling is another sign that increasing levels of carbon dioxide are changing our planet's climate.
Introduction to Ozone (UCAR Education & Outreach)
Repairing the Antarctic Ozone Hole (Windows to the Universe)
Ozone Depletion (U.S. EPA)