Is it possible for me to derive Avogadro's number?
To determine Avogadro's number you have to measure the same unit at the atomic and macroscopic scales. 

This was first achieved by Millikan who measured the charge of an electron. The charge of one mole of electrons was already known and  is a Faraday. Dividing both, you get Avogadro's number. 

Before that, Josef Loschmidt was the first one to calculate the number of particles in a cubic meter of gas, using the kinetic theory of gases. 
This is, of course, also related to Avogadro's number. 
Avogadro himself did not give any number. He just stated that equal volumes of different gases at the same pressure and temperature have equal amount of particles. 

A more modern way of doing it is determining the density of an ultra pure element and then determine the number of atoms and their distances in a unit cell with X-ray diffraction. 

So yes, you could, in principle, derive Avogadro's number with pure monocrystalline carbon 12. You will, however, need very good instrumentation. 

As per the answer to this question on Chemistry SE it might become the other way round. 
The Avogadro number will be absolute (defined, with 0 error) and then the kilogram will be redefined as a function of the number of atoms in a monocrystalline perfect $\ce{^{28}Si}$ sphere.