Amplify the signal from ACS712 to adc
The output from the ACS712 is centred at 2.5 volts dc and, for a range of input currents will produce an output waveform that can peak up at 3.5 volts or trough down as low as 1.5 volts.

You have an ADC that requires it to be centred at 2.5V so half the battle is done. I would strongly consider doing nothing and let the ADC/MCU convert the sampled current into an RMS value. This, of course, means taking continual samples at several hundred hertz in order to properly measure the RMS current and avoid a lot of aliasing.


  what I want to take measure are things like energy-saving lights, or
  other things with low consumption.


What you propose in your circuit is a very basic rectifier circuit and this immediately introduces a "several-percent" error in measurement and also restricts your ability to only measuring sinewaves and not more complex current waveforms (that are typical in building and household appliances).

To make your circuit work better you need a bleed resistor across C1 but, I urge you to do all signal processing using an MCU and forget about adding diodes and capacitors externally.

If you want to measure power you need to sample voltage and current at a rate not less than 1kHz to get any measure of accuracy.