This is p.205 of Ta-pei cheng book. The dual field tensor is defined in $(10.52)$ in terms of the covariant tensor. Then I think the definition of contravariant dual field tensor is just reversing the indices locations from $(10.52)$. Am I correct? If so, how can I derive the definition from $(10.52)$?
If you are asking about $\tilde{F}^{\mu \nu}$ then you can compute it in any of the following forms $\tilde{F}^{\mu \nu} = g^{\mu \alpha} g^{\nu \beta} \tilde{F}_{\alpha \beta} = - \frac{1}{2} \varepsilon_{\alpha \beta \gamma \delta} g^{\mu \alpha} g^{\nu \beta} F^{\gamma \delta} = - \frac{1}{2} \varepsilon_{\alpha \beta \gamma \delta} g^{\mu \alpha} g^{\nu \beta} g^{\sigma \gamma} g^{\tau \delta} F_{\sigma \tau} = - \frac{1}{2} \varepsilon^{\mu \nu \sigma \tau} F_{\sigma \tau} \, $. Be careful about the fact that in terms of numerical value $\varepsilon^{\mu \nu \sigma \tau} = - \varepsilon_{\mu \nu \sigma \tau} $.
