Plug in eq. 6 into eq. 5 and solve carefully. For the solution to hold for all x and t, the cosine terms must cancel each other, and the sine terms must cancel each other. This gives two constraints involving kappa, k, omega, omega_0 and c_0. Find those constraints as two equations.
Assumed in eq. 6 is that the solution is physical, so that the amplitude decreases in the direction of the wave propagation. This means that kappa and k must have the same sign. Without loss of generality, assume k>0 and kappa>0. Use one constraint to show that we then have kappa < k, and furthermore that 2*kappa is approximately k*omega/omega_0 when omega << omega_0.
Combine this approximate result with the other, so far unused, constraint to eliminate k and find kappa as a function of omega, omega_0 and c_0.