Translating voltage using op-amp
Amongst the problems you have are getting high gain in a single stage, DC offsets of the amplifier (which for cheap ones exceed the input voltage range), and making sure the amplifier can cope with the negative input.

You say nothing about the output impedance of the sensor, or the speed you want, or whether you want to stick to single rail opamps or can cope with dual rail, so some assumptions must be made.

The two simplest toplogies are inverting, and non-inverting, below







simulate this circuit &ndash; Schematic created using CircuitLab

The inverting topology uses R3 and R4 to provide a gain of 5000, to boost your 1mV range up to 5v. Most amplifiers will have enough gain to do this.

It assumes the amplifier has an input common mode range that includes ground, and an output that will swing 0 to 5v. This can either be a rail-to-rail amplifier with a 0/5v supply, or any amplifier with appropriate power rails.

Most cheap amplifiers will have an offset voltage in the several mV, so it's a nonsense to attempt to set the input voltage tighter than this. What we now need to do is adjust the input offset until the output voltage falls in our 5v range. I've shown a current source of 1uA. If this is connected to R1, it will lift the voltage of the +ve input by 1mV, increasing the output by 5v. If it is connected to the R3/R4 junction (the summing junction) then it will reduce the output by 5v. Ajusting the value of this current source, which need only be a high value resistor from the 5v supply, and choosing which point it's connected to, will allow you to offset the amplifier's offset error, and accomodate the DC input shift from your sensor.

The non-inverting topology to the right has much the same considerations. It offers a high input impedance to the sensor. As before, the 1mV range of the sensor is swamped by the input offset of the amplifier. You can connect an offsetting current into the R5/R6 junction, but you need to be able to swing this current both positive and negative if you want control of offset voltage in both directions.

You can obtain some single op-amps with input voltage offset adjust pins. I don't know whether any of these versions are available in 5v rail-to-rail. Certainly TL071 has offset adjust, but you would need rails of at least +10v and -5v to make it cover the range.

You would need to choose an amplifier with a low tempco of input offset voltage, if your carefully set up input offset was not to drift around with temperature. For instance, the LM324 (popular cheap low voltage amplifier, but not rail to rail!) has a 7uV/C input tempco. A change in temperature of 5 degrees C could look like an input signal of 3.5% of full scale! There are much better ones, OP177 for instance (0.3uV/C).

You can get chopper stabilised opamps, which have very stable DC offsets, and are not ruinously expensive.

You might want to look at in-amps, instrumentation amplifiers, and see how they improve of some of the limitations of single amplifiers.