The LM35 has 10 mV/Â°C output, without any help the 10bit ADC (with 5V reference) can read 5/0.01 = 500 Â°C (LM35 melt before) with 500/1023 = 0.488 Â°C resolution.
The LM35 can work till 150Â°C, so you can increase the resolution by reducing the reference or amplifier the signal. The circuit with OP use the second way. To read it you do the same, just take care in the calculation that the value is not 10 mV/Â°C, but higher.
If the OP amplify x4 you have 40mV/Â°C = full scale at 125 Â°C, 0.1222 Â°C/bit resolution. The Â°C display out is = ADC read x 0.1222.
But I think there is a bettor and simpler solution, with much higer resolution.
Some year ago I built a multi point thermometer, and I found that a very good solution is the LM92 from National. It has hight resolution, is DIGITAL, no analog, non OP, no reference, no noise, just read it by I2 bus. The interface is very simple.
The IC is a SMD device, so put a SMD 0.1 uF capacitor directly on IC's pins.
In my solution the sensor was glued with thermo plastic tube and a 4 wire small cable.http://cache.national.com/ds/LM/LM92.pdf