Need to alter a 3 V - 4.2 V output to a 1 V - 5 V input
You need amplification since:
$$\frac{5-1}{4.2-3} = 3,333$$
So you could use a differential amplifier:

to get the desired result:

V1 is the voltage you want to read. V2 shifts the output up and V3 is there to match the minimum input voltage. Both V2 and V3 can be provided by simple voltage references like a TL431 and you may need precision resistors to avoid using trimpots to match the resistor pairs (R1 = R5 and R4 = R6).
Edit: assuming a bargraph with 10 steps, a \$(5-1) / 10 / 2 = 200 mV \$ error would be acceptable. A random simulation with 10000 runs using 1% resistors showed a maximum error of 150 mV.