Derivation of the van 't Hoff formula

The equation linking ΔH∘ΔH∘ and KK is called the van 't Hoff equation. Since Philipp's comment on your question already links to a very thorough discussion of where the equation ΔG∘=−RTlnKΔG∘=−RTln⁡K comes from, I won't repeat it.

The definition of the Gibbs free energy, GG, is G=H−TSG=H−TS; if the temperature of the system is kept constant, then

ΔG=G2−G1=(H2−TS2)−(H1−TS1)=(H2−H1)−T(S2−S1)=ΔH−TΔSΔG=G2−G1=(H2−TS2)−(H1−TS1)=(H2−H1)−T(S2−S1)=ΔH−TΔS

and if we impose the standard state (note that standard state does not specify a temperature!) on all species present, then we get

ΔG∘=ΔH∘−TΔS∘ΔG∘=ΔH∘−TΔS∘

So, equating our two expressions for ΔG∘ΔG∘, we get:

ΔH∘−TΔS∘=−RTlnKΔH∘−TΔS∘=−RTln⁡K

lnK=−ΔH∘RT+ΔS∘Rln⁡K=−ΔH∘RT+ΔS∘R

Here, ΔH∘ΔH∘ and ΔS∘ΔS∘ are constants at any single temperature. If we now differentiate both sides with respect to temperature, we obtain

d(lnK)dT=ΔH∘RT2d(ln⁡K)dT=ΔH∘RT2

This is the differential form of the van 't Hoff equation; it's not the most useful thing to us though because it only tells you the slope of a plot of lnKln⁡K against TTat one given point. We usually separate the variables and integrate with respect to both sides:

∫lnK2lnK1d(lnK)=∫T2T1ΔH∘RT2dT∫ln⁡K1ln⁡
K2d(ln⁡K)=∫T1T2ΔH∘RT2dT

lnK2−lnK1=ΔH∘R(1T1−1T2)ln⁡K2−ln⁡K1=ΔH∘R(1T1−1T2)

So, if you know the equilibrium constant K1K1 at a certain temperature T1T1 and you want to find the equilibrium constant K2K2at a different temperature T2T2, you can just plug in your values into the equation and solve for K2K2.

Note that this equation supports what you know of Le Chatelier's principle; if the reaction is exothermic, ΔH∘<0ΔH∘<0, and if you increase the temperature from T1T1 to T2>T1T2>T1 
then (1/T1−1/T2)>0(1/T1−1/T2)>0. The RHS of the equation is therefore negative, and that means that 
lnK2<lnK1⇒
K2<K1ln⁡K2<ln⁡K1⇒K2<K1 which implies that the equilibrium position has shifted to the left.

Note that the last step (the integration) makes the assumption that ΔH∘ΔH∘ is a constant over the temperature range T1T1to T2T2. Note that this is, in general, not true but if the temperature range isn't too huge you will get pretty accurate results from the use of this equation