You'll need a fixed voltage reference to compare the battery voltage to, and to drive LEDs you'll likely need a boost converter otherwise if the battery is below 2V or so the LEDs won't light.
If you use a boost-mode switching regulator it can also serve as the reference voltage source. If your battery voltage is 1-3V that's a 2V range, with 6 levels to measure (assuming the lowest is no LEDs lit and the highest is all 5 lit). That gives you 0.33V per LED. So, you'd want the 1st LED to light at 1.33V, the 2nd at 1.66V, the 3rd at 2V, the 4th at 2.33V, and the 5th at 2.66V. A resistor ladder can be used to derive the reference voltages into the inverting inputs of 5 comparators; then feed the battery voltage directly into the non-inverting inputs of the comparators. Connect the LEDs to the comparator outputs with current-limiting resistors.
I'll leave it as an exercise to figure out what the resistor values in the ladder need to be. Partly because my algebra is a bit rusty after 30-odd years.
