We present a systematic investigation of two-photon excitation processes in a GaAs-based microcavity in the strong-coupling regime. We observe second harmonic generation resonant to the upper and lower polariton level, which exhibits a strong dependence on the photonic fraction of the corresponding polariton. In addition we have performed two-photon excitation spectroscopy to identify $2p$ exciton states which are crucial for the operation as a terahertz lasing device, which was suggested recently [A. V. Kavokin et al., Phys. Rev. Lett. textbf{108}, 197401 (2012)]. However, no distinct signatures of a $2p$ exciton state could be identified, which indicates a low two-photon pumping efficiency.
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