How is hydrostatic pressure overcome when a star is formed?

The answer lies in something called the virial theorem.

You are correct, a cloud that is in equilibrium will have a relationship between the temperature and pressure in its interior and the gravitational "weight" pressing inwards. This relationship is encapsulated in the virial theorem, which says (ignoring complications like rotation and magnetic fields) that twice the summed kinetic energy of particles ($K$) in the gas plus the (negative) gravitational potential energy ($\Omega$) equals zero. $$ 2K + \Omega = 0$$

Now you can write down the total energy of the cloud as $$ E_{tot} = K + \Omega$$ and hence from the virial theorem that $$E_{tot} = \frac{\Omega}{2},$$ which is negative.

If we now remove energy from the system, by allowing the gas to radiate away energy, such that $\Delta E_{tot}$ is negative, then we see that $$\Delta E_{tot} = \frac{1}{2} \Delta \Omega$$

So $\Omega$ becomes more negative - which is another way of saying that the star is attaining a more collapsed configuration.

Oddly, at the same time, we can use the virial theorem to see that $$ \Delta K = -\frac{1}{2} \Delta \Omega = -\Delta E_{tot}$$ is positive. i.e. the kinetic energies of particles in the gas (and hence their temperatures) actually become hotter. In other words, the gas has a negative heat capacity. Because the temperatures and densities are becoming higher, the interior pressure increases and may be able to support a more condensed configuration. However, if the radiative losses continue, then so does the collapse.

This process is ultimately arrested in a star by the onset of nuclear fusion.

So the key point is that collapse inevitably proceeds if energy escapes from the protostar. But warm gas radiates. The efficiency with which it does so varies with temperature and composition and is done predominantly in the infrared and sub-mm parts of the spectrum - through molecular vibrational and rotational transitions. The infrared luminosities of protostars suggest this collapse takes place on an initial timescale shorter than a million years.

As gas clouds collapse, they increase in internal energy (measured by temperature). This is part of what causes their pressure to increase. As they increase in temperature, though, they also increase the amount of radiation they emit. As they emit radiation, their internal energy decreases and thus their pressure also decreases, allowing for further collapse.

This makes it seem like the temperature would decrease, but it turns out that the physics involved makes it so that when internal energy is removed from a collapsing gas cloud (by radiating the energy away), the amount of gravitational energy that is converted to internal energy is greater than the amount of internal energy that is radiated away. Thus, the temperature increases as the gas cloud collapses, even though the collapse is happening because energy is radiating away. In this way, collapsing gas clouds can be considered to have a negative specific heat: removing energy from the system actually increases the temperature, because of the gravitational energy that is released.

For a uniform, spherical distribution of mass (cloud of gas and dust) of radius $R$ and mass $M$ in absence of magnetic, radiation fields etc, we have $dm = 4\pi \rho r^2 dr$ and the potential energy of a spherical shell of inner radius $r$ and outer $r + d r$ is $dU = -G\frac{m(r)dm}{r}$, $m(r) = \frac{4}{3}\rho r^3$, and a simple integration yields, $$U(r) = -\frac{3}{5} \frac{GM^2}{R}.$$ For a mass distribution to collapse, according to the virial theorem, $2K + U <0$, with $K$ the kinetic energy of the mass, mainly due to thermal motion, i.e. $K = \frac{3}{2} NkT$, where $N$ is the number of molecules. If we express the mass of the cloud as $M = N \mu$ with $\mu$ the mean mass of the molecules, then the previous inequality for the cloud to collapse becomes $$M> M_{\text{Jeans}}, \quad M_{\text{Jeans}} \equiv \left( \frac{5kT}{G\mu} \right)^{3/2} \left( \frac{3}{4\pi \rho} \right)^{1/2},$$

the condition involving the Jeans limit for the mass of a cloud. It is obvious from this inequality that dense and cold clouds are easier to collapse and form a star.

The equation that relates the pressure and the density is known as the equation of state; it is in general unknown.