How to wire an AC induction motor
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Single-phase capacitor start induction motor.

Let's just get the thing running first.

Add the capacitor in series with the winding with higher resistance, if there is one. If the motor runs backwards then reverse one of the windings.

Explanation

If we only had Winding1 and apply an AC voltage to it the motor will run if you spin-start it by hand (very carefully). The problem is that it doesn't know which way to turn.

Winding2 is offset slightly - maybe 10° or 20° - and the capacitor causes the current to lead that in Winding1 enough that it pulls the rotor slightly ahead of Winding2 giving enough rotation that the motor starts reliably and with the desired direction.