How can I tell my dell precision 7510 battery is dead?

Actually you can easily tell acording to the battery charging light color.

Alternately blinking amber light and white light
    An unauthenticated or unsupported non-Dell AC adapter is attached to your laptop. Re-plug battery connector, replace battery if the issue reoccurs.

Alternately blinking amber light with steady white light
    Temporary battery failure with AC adapter present. Re-plug battery connector, replace battery if the issue reoccurs.

Constantly blinking amber light
    Fatal battery failure with AC adapter present. Fetal battery, replace the battery.

Light off
    Battery in full charge mode with AC adapter present.

White light on
    Battery in charge mode with AC adapter present. 

The above source from official manual.

dell precision 7510 battery

The brand new original dell precision 7510 battery, dell precision 7510 11.4V 7950mAh laptop batteries detail is as below:

    Voltage: 11.4V
    Capacity: 7950mAh (91Wh)
    Battery cells: 9-cell
    Chemical: Li-ion
    Battery Color: black
    Condition: new original battery
    Warranty: 9 months

The battery is in stock and available here: https://www.storebattery.co.nz/dell-precision-7510-11.4v-7950mah-laptop-battery-store-for-new-zealand-p-174361.html