Does multiplying range affect accuracy of an analog to digital converter?
The resolution in volts is the reference voltage of the analog to digital converter (ADC) divided by the number of counts from the ADC.
The input voltage of the signal isn't involved.
Given a 10 bit ADC (1024 counts) and a 5V reference voltage for the ADC, the voltage resolution is 5V/1024 = 4.8828125 millivolts per count.
Your input signal of maximum 1V would generate a maximum count of 204 from the ADC.
If you need 1V/1024 resolution, then you have a couple of options:

Use an opamp to multiply the input signal by 5 before it goes to the ADC
Use a reference voltage of 1V for the ADC.

Both options have disadvantages.
For example, the opamp solution can add noise and offset to your signal.  Using a lower reference voltage depends on having a clean enough (and accurate enough) reference voltage available - and your ADC must have a connection for an external reference.