The Koopman operator approximation is emerging as a leading approach for identifying and controlling non-linear systems by transforming them into a bilinear form. However, designing reactive controllers for the Koopman bilinear system remains challenging. This paper proposes a purely data-driven method to learn the Koopman bilinear representation of control-affine systems using measurement data only and design a linear optimal controller for the learned system. Specifically, Deep Neural Networks (DNNs) are employed to learn a finite set of observables that approximately span a Koopman-invariant subspace and form a multiplication-closed set. This multiplication-closed property facilitates optimal controller design by enabling the conversion of the Koopman bilinear system into a closed-loop linear system. The linear control matrix is derived by iteratively solving the Koopman Riccati equation while minimizing an upper bound of the optimal cost. The proposed approach is validated on the Van der Pol oscillator, which outperforms the method that approximates the Koopman control system using a fixed function library in prediction accuracy and control performance.