Could Calcium Oxide be made from CaCO3 in a bonfire?
According to wikipedia, CaCO3 will turn into Calcium if heated above 825 °C (1,517 °F).

As for your fuel 

Wood burns in several phases.

In the first phase, which can be referred to as dehydration, the heat produced by the wood's combustion drives of water from it in the form of vapour. It takes a lot of energy to do that which limits the fire's temperature to ~200-250°C.

The second phase, which may be referred to as pyrolysis, kicks in as the wood gradually becomes dehydrated. In that phase the fire's temperature will go from 250°C to topical peaks up to ~800°C.

As the wood further breaks down, you will encounter a third phase which is equivalent to the burning of charcoal. The temperature at the heart of your ambers will be at a minimum of ~700°C and could reach as much as 1300°C if you bring enough oxygen into it (by injecting compressed air into the heart of your ambers through a metal pipe for instance). This temperature would be fully sufficient to produce CaO from CaCO3 in a reasonably short amount of time.

At this point you could also add regular coal into the ambers, which requires a temperature above ~700°C to start burning. In ideal conditions, coal can burn up to ~ 2100°C which would significantly push the limits of what is possible for you.

If you would like to push these limits further, you could add coke on top of your regular coal, which would push the theoretical maximum temperature your fire could reach up to ~ 2500°C.

More practically

As Francis mentioned, there is a wide gap between theory in ideal conditions and practical reality.

The first aspect I already mentioned that will affect the practical temperature your chalk will be exposed to, is the amount of oxidising molecules that is brought to the heart of your fire. Injecting air through a pipe may be the safest option (though it is already significantly less safe than just making a regular bonfire, as it will significantly increase the combustion temperature and is at risk of propulsing fuel around if not adequately set up). 
Theoretically, you could achieve a yet better result (at the cost of a lot of danger) by injecting pure compressed O2 into the heart of your ambers (which, for safety reasons, you should NOT do).

The second and very important aspect is insulation. The theoretical maximum temperature that is reached within your fire is of little practical importance. Insulating your fuel combustion zone without compromising the supply of oxidisers into it is key not only to increasing the maximum temperature that is locally reached inside of your fire, but more importantly to just keep calories inside of the combustion zone, which will significantly increase the average temperature inside of it (and tremendously lessen the required fuel expense for achieving any given amount of CaCO3 calcination). You can insulate your fire in numerous ways. Coal itself can act as a (burning) insulator if you place your CaCO3 at the heart of your combustion zone and cover it in a pile of coal/charcoal. Or you could build a structure around your fire: SiO2, Al2O3, CaO and MgO are particularly suitable materials for their resistance to high temperatures and for their refractive properties. You can look up their properties on wikipedia (MgO for instance melts at 2,852 °C, has a thermal Conductivity 45–60 W·m−1·K−1 and a Refractive Index of 1.7355).

Chemistry

My last point is that there might be the possibility for you to make things easier by adding some additional chemistry into the process (if you are inclined to do so). It would seem, for instance that Ca(OH)2 might turn into CaO starting around 512°C, which seems to be lower than CaCO3. So you might be able to lower your requirements on the combustion side by turning your CaCO3 into Ca(OH)2 prior to calcination. There are many ways to do this but one way would be to react CaCO3 with HCl to yield CaCl2 , which you'd subsequently react with NaOH (the very low solubility of Ca(OH)2 would drive the reaction forwards).