There are three main approaches to prove the Black Scholes formula. They include:
1) The PDE Approach (you transform the problem to a heat equation PDE and solve)
2) The Stochastic DE approach (You use Ito's result and solve)
3) The Probability Approach (You find the random variable that corresponds to a Price at time t and then take the expectation to get the fair price)
All approaches will give the same answer and each requires a different kind of thinking and reasoning.