It would be unusual for an ADC to cover positive and negative voltages, nor is it required. You simply bias the input of the opamp to +2.5V, a negative voltage will move it down towards zero, and a positive value will move it towards +5V.
It's a simple circuit, you need the output of an opamp that can move between -5V and +5V (10V range), when it's input is zero, then the output should be zero as well. Connect the output of the opamp via a resistor to the input of the A2D, and a second resistor from the input of the A2D to 5V, the two resistor should be identical. Notice that the two resistors do two tasks, they shift a negative to positive swing to a zero to positive one, and also divide by two - converting the 10V swing to a 5V one.
Your software needs to take care of the rest, it's quite straight forward to do.