Recording bioelectrical activity in plants- will this circuit work- any advice on sensors?
Here are a few problems you are going to run into:


Your circuit will output a voltage that ranges between +7 volts and -7 volts.  The Arduino analog input can accept voltages between 0 volts and +5 volts.
The charge pump you built around the 555 will be electrically "noisy."  It will produces interference in the signal you are trying to measure, making your task much more difficult.
The 741 is not known for high performance.  It was a wonderful part back in 1963 when it was developed, but it has been surpassed by just about everything produced in the mean time.  It will cause you all kinds of difficulties in measuring very weak signals.
The input impedance of your proposed circuit is likely far too low. This paper mentions measuring millivolt level signals using a commercially available voltmeter with an input impedance in the gigaohm range.  Your circuit will probably have an input impedance of several hundred kiloohms.


You need to do a few things before you get started:


Determine just what a biolectrical signal looks like.  


Is it DC?
If so, how do you determine the polarity?
If it is AC, what frequency range can you expect?
Regardless whether it is AC or DC, what voltage range can you expect?



That all determines what you are going to have to do to get a signal that the analog to digital converter of the Arduino can work with.  That's gain (amplification) and level shifting (will you have to move negative voltages up above zero) and bandwidth of the amplifier.


Given the very weak signals, it might be better to use a second battery instead of a charge pump to generate a negative voltage source.
You need an instrumentation amplifier with high impedance inputs.  You won't be getting that from an in-amp assembled from the 741.  Use a proper in-amp and save yourself a lot of hassle.


There are a lot more details you'll have to handle.  The points I listed are just a starting point.  You've picked a difficult task, and don't have enough details to really plan things yet.  You are going to have to gather details and knowledge about the subject.  As you progress, I expect you'll find things you didn't know you needed to consider.  Come back and ask specific questions about those things.