Increasing voltage maximum value from 10 mv to 5v
A solution is to amplify the 0 to 10 mV using an opamp circuit, here's an example: 

I would also increase the value of the shunt resistor though, 10 mV full range is not much. 100 mV would give you more immunity to noise and other disturbances. The power dissipation in the shunt resistor will increase though.

Another solution could be a small module like this sensor
it has an the added benefit, the side where the current flows is isolated from the part which you need to connect to the ADC. Check the specification if the accuracy is good enough for your purpose !