Using a simple excel spreadsheet and some assumptions:
Vout/Vin = 5V/0.1V = gain of 50
Thus (1+R4/R3) = 50, R4=500 & R3=10 yields gain of 51, close enough
Plugging in steady state V1 = 0V, V2 = 2.5V/50, output voltage of 2.5V, R3 & R4 above,
yields R2 = 9 * R1. 100K & 10K are close enough
R1 = 10K
R2 = 100K
R3 = 10K
R4 = 500K
V2 = 0.05V
For V1 = 0.01V, Vout = 0.695V
For V1 - 0.1V, Vout = 4.868V
Similar results should be achievable.
If your op amp won't do that full of a voltage range, lower the gain and increase the offset:
V2 = 0.1V, R4=400K:
V1 = 0.01 -> Vout = 0.745
V1 = 0.1 -> Vout = 4.1V