How can I build a receiver to draw power from an alternating electrical field?
A typical EEG would measure signals in the range of about 10 µV to 100 µV in amplitude when measured from the scalp[31] and is about 10–20 mV when measured from subdural electrodes
 and the frequency is well below 20 KHz, and frequently below 70 Hz.  

This paper by Linear Tech talks about harvesting energy from sources as low as 20 mV.  So, in theory, the technology is barely able to extract energy from electrodes placed under the skin.  And by "just barely", I mean that it might work in a lab but that's about it.  It won't work from electrodes placed on the skin.

What we don't know is just how much power (watts) could be extracted.  It it probable that there isn't enough to power a very efficient LED or MCU.  

Keep in mind that if you are efficient at extracting energy then you are in effect preventing the energy in that "brain signal" from going to where it is needed-- in essence, preventing your brain from doing its function.  I do not see this as a viable source of energy.

There are other ways to harvest energy from vibrations and shock-- and theses should be more useful and practical than what you are proposing here.  A vibration sensor placed on the hands/feet or even over an artery on your neck/arm/leg would give you more energy (educated guess) than inserting an electrode under your skin.  And it wouldn't turn the world into a real-life version of The Matrix.