You are a little confused in your stellar evolution model. After the ignition of hydrogen fusion in the core of a star, it will next progress to helium fusion, then to carbon/oxygen fusion via the triple-alpha process (I've skipped a lot of steps and details there, if you want the details you can look at either Hansen & Kawaler's Stellar Interiors text or Dina Prialnik's Introduction to Stellar Structure text). What happens next is mass-dependent (using $M_\odot\simeq2\cdot10^{33}$ g and the mass of the star as $M_\star$):
- $M_\star\gtrsim 8M_\odot$
- able to continue fusion in the core
- will later blow up in core-collapse supernova events, producing either a neutron star or a black hole (mass-dependent) after forming iron in the core
- $M_\star\in(\sim0.5,\,\sim8)M_\odot$
- unable to continue fusion in the core due to insufficient temperatures
- will proceed into the planetary nebula phase (which has nothing to do with forming planets, but it's discoverer, William Herschel, thought that it was a planetary system forming)
- these stars form the white dwarfs that the Chandrasekhar limit applies to
- $M_\star\lesssim0.5M_\odot$
- unable to produce helium in the core (insufficient temperatures)
- expected to continue burning hydrogen for $t_{burn}>t_{age\,of\,universe}$
Thus, not every star produces iron in the core; this only applies to stars with mass $\gtrsim8M_\odot$.
The Chandrasekhar limit arises from comparing the gravitational forces to an $n=3$ polytrope (see this nice tool from Dr Bradley Meyer at Clemson University on polytropes)--polytropes basically mean $P=k\rho^{\gamma}$ where $P$ is the pressure, $k$ some constant, $\rho$ the mass density and $\gamma$ the adiabatic index.
That is, in order to find the limit, you need to use the hydrostatic pressure,
$$
4\pi r^3P=\frac32\frac{GM^2}{r}\tag{1}
$$
and insert the pressure of the polytrope of index $n=3$ (requires numerically solving the Lane-Emden equation) and then solving (1) for the mass, $M$. If you've done it correctly, you'll find $M_{ch}=1.44M_\odot$.