Think of a basic RC circuit where the R and C are in parallel. Our goal is to have a clock at the output of this circuit - a 0-5V 1KHz square wave. So when we want the clock to be high we turn on our voltage source and it charges up the capacitor until the output is at 5V, and when we want 0V we turn it off and let it discharge. The charge/discharge time is determined by the RC constant of the circuit. There's a problem - the circuit doesn't charge up quick enough for a 1KHz clock. What do I do?
We can't change the RC constant of the circuit - it's fixed. So we have to charge the capacitor up quicker somehow, but still have the same charged voltage. To do this we need an active circuit that monitors the output voltage of the RC circuit and varies the current going into the capacitor to charge it up quicker. More current means more power.
When you want a faster clock, you need to charge up the capacitor faster. You charge up a capacitor by pushing current into it. Current * voltage = power. You need more power!
Everything in a digital system is tied to the clock and everything has capacitance. If you have 100 TTL chips on one clock it has to drive a lot of current to charge all of them, then draw a lot of current to pull them down. The fundamental reason ohms law isn't holding is because these are active devices, not passive. They do electrical work to force the clock to be as close to a perfect square wave as possible.
If you overclock a microcontroller it gets hot
Yes - quicker change means more current flowing and power is voltage * current. Even if voltage stays the same, current used increases, so more power dissipation, more heat.
If you overclock a microcontroller it needs more voltage
Partially true - it needs more power, not necessarily more voltage. The microcontroller is in some way converting the extra voltage to more current to achieve its needs.
As far as I know, the frequency of AC has nothing to do with its voltage or power, and a clock is just a super-position of a DC and a (square) AC. Frequency doesn't affect the DC.
Only for a purely resistive load. There's a lot of trickery happening with AC power.
Is there some equation relating clock frequency and voltage or clock frequency and power?
Probably not a consistent one, but it's related to the simple equations Q=CV, V=I*R, P=I*V
Just remember: Higher frequency => faster rise time => must fill up capacitors quicker => more charge => more current => more power.