how to access iPad battery cycle counts?

You're most likely going to have to be jailbroken, but iBackUpBot is your only option.

You can have the Apple Store check your device out, although they may not tell you what the capacity and your numbers are.

There are several tweaks through Cydia, such as BatteryDetective and BatteryInfo for Notification Center.

Some specifications are listed:

- current charge (mAh)
- maximum charge (mAh)
- battery percentage (rounded to two decimal places)
- battery state (discharging, charging, or fully charged)
- current capacity (mAh)
- design capacity (mAh)
- battery health (rounded to two decimal places)
- temperature of the battery (to two decimal places)
- voltage of the battery
- time remaining (until fully discharged or fully charged)
- cycle count (number of times you discharged and charged your device)

If not jailbroken, open iBackUpBot
If jailbroken, open Cydia

The cycle count feature only works on the iPhone 3GS, iPhone 4, iPhone 4S, iPhone 5, iPad, and the iPad mini.

I'd recommend jailbreaking, if applicable, and give the tweaks a try.