What energy density is required for the batteries in order to make an all-electric analogue of the Cessna 150 or similar plane?
Short Answer

Batteries would need to be somewhere around 16.7MJ/kg to give the same range and performance as liquid fuels, this is about 18.5 times the capacity of the best lithium-ion batteries. Price-wise it will cost about 30-35% to charge your airplane as opposed to fill it with liquid fuels at today's prices.  

Long Answer

This is a good question which is hard to give an exact answer to, so this will be more of a Fermi Approximation. To answer this you need to look at the energy contained in the fuel and the efficiency of the engine used. 

Looking at the energy of the fuel I will use Specific Energy, which is the energy stored in a material per unit of mass. Specific Energy is related to Energy Density, which is the amount of energy contained per unit of volume. Often the terms are interchanged. 

The specific energy of avgas and jet fuel is about 43 MJ/kg. The best lithium-ion batteries top out at about 0.9 MJ/kg (the batteries in a Tesla are about 0.7 MJ/kg), so they have a fraction of the storage of liquid fuels. The best battery technology in theoretical development (Lithium-air) has a theoretical maximum of 41 MJ/kg, more realistically they'll get 1/4 to 1/3 of that from the technology, which is still vast. 

Internal combustion engines are about 35% efficient, the other 65% is wasted whereas electric engines are much closer to 90% efficient or more.

There are other factors to consider:


Weight: fuel tanks, piping and pumps take weight, and electric motors are much lighter than internal combustion engines. The batteries would need a storage structure but an electric system would probably be lighter overall
Battery inconsistencies: avgas supplies a consistent amount of energy in every single drop whereas batteries will sag as they discharge (this is regulated to give consistent power to the motor, however at some point levels will drop below what is regulatable, so not all the power in the batteries will be usable). Also, over time they lose capacity and become less efficient. Both these factors mean that you'd want to build in extra battery capacity to compensate


So I'm going to assume these two will compensate for each other, the weight savings from going electric will be offset by the need for extra capacity to ensure consistency. Assuming that all other factors are equal (propeller efficiency, etc) we can extrapolate the actual efficiency of the systems to get an approximate figure of an internal combustion on avgas: 35% of 43 MJ/kg = 15 MJ/kg of actual benefit. We can use that figure to determine what specific energy we would need from batteries to get the same amount by dividing by electric motor efficiency: 15 MJ/kg / 0.9 we get 16.7 MJ/kg. 

So batteries would need to store 16.7 MJ/kg to give us the same energy as liquid fuels, but how does it compare to existing battery tech? Right now commercially available battery technology is about 0.9 MJ/kg, so it would need 18.56 times more storage capacity (16.7 / 0.9 = 18.56) to supply the same amount of energy. 

As for energy costs this is going to vary a lot depending on fuel and electricity prices over time and place; we can use some of the same figures above to work the numbers. I'll assume an airplane that holds 40 US Gallons (150 liters) as the math is easy and it's about the capacity of a Cessna 172. I'm going to do separate calculations for the US and UK to see how they compare:


In the US: Avgas is about \$5.20 per US gallon at the moment, that's \$208 to fill up your airplane of 40 gals. Avgas has about 34.2 MJ/liter of energy, multiply that by 150 liters for 5130 MJ of energy. Internal combustion engines are 35% efficient and electric engines 90%, so we can calculate that an electric airplane would need 5130 * 0.35 / 0.9 =  1995 MJ of electrical power to fill up. 1995 MJ is about 554 kWh, at somewhere around \$0.135 per kWh it would cost you \$75 to charge an airplane with the same amount of energy.
In the UK: Avgas in the UK is about \$2.23 per liter (\$8.47 per US Gallon (ouch!)), so it would cost \$334.50 to fill up an airplane. 554 kWh of electricity costs about \$0.17 per kWh, so it would cost \$94 to charge the airplane


EDIT:
I did consider the efficiency gain from weight loss as fuel is burned, i.e. a variable mass system, I left it out of the answer as it isn't significant compared to the other factors in what is already an approximation. It has to do with the fuel fraction, which is the percentage of the aircraft weight that is fuel, which on a light piston single is comparatively low. A Cessna 172 carries about 40 gallons of fuel, 38 of which is usable, weighing about 228 pounds as opposed to a typical take-off weight of 2200-2300 lbs. In other words its fuel fraction is about 10%, even on a long range flight using every bit of fuel you will only lose 10% of your weight, and I would approximate that you'd get around 5% benefit from it. This was not enough of a factor compared to the other considerations, and it would probably be offset by regenerative descents where the electric motor actually generates power from the spinning prop. 

For airplanes where a more significant portion of the weight is fuel, i.e. a higher fuel fraction, efficiency gains from variable mass is a much more significant factor, for example an A380's fuel fraction is 44%.