Measure capacitance with OPA604 and this circuit
How have you assembled it?  On a PCB, plug-in breadboard or overboard.

The gain at 50/60Hz will be very large and it is easy to get coupling from the AC power lines. I would suggest placing a grounded metallic plate under the circuit - make sure you insulate it appropriately. Attach that to the circuit ground rail.

To isolate the low frequency interference disable the 100kHz signal.

As others have said the 1n4007 is not at all appropriate, use a 1n4148 or similar signal diode.  You will not get good accuracy with just a simple rectifier, it will be very non-linear.  You will probably need to use a synchronous rectifier, you can use an CD4053 or similar to create that.

The ripple at the output is just because you have a large amount of low frequency interference - the time constant of the 1000pF cap and the scope probe input resistance is such that it follows the interference - get rid of that and it should be much better.