As Cristian already pointed out...the input to the chip can be only 0->5v if you tie Vref+ to +5v and Vref- to GND. This applies to ANYTIME, not just to power up time. Thus you need to come up with an opamp circuit that accepts an input voltage range of -5v -> +5v and translates that range to 0 -> 5v.

The converter (if a 10-bit full resolution) will then nominally produce a results as follows:

Input  A/D Input   A/D
Volts    Volts    Result
-----  ---------  ------
+5.0     +5.00     1023
+2.5     +3.75      768
 0.0     +2.50      512
-2.5     +1.25      255
-5.0      0.00        0

With regard to the display issue, it becomes a software project to scale the A/D readings that you get to the corresponding display values that they represnt. In the above example ano value you see above 512 (i.e. MSB of A/D reading set) could be taken as a positive value. So you would take:

Display Value = ((A/D Reading - 512) * 5)/512

Alternatively if the msb was clear on a reading that is taken to correspond to a negative input value so you would take:

Display a minus sign then:
Display value = ((512 - A/D Reading) * 5)/512

Good Luck
Michael Karas