the second one is in degree 0. Our strategy to compute these groups is to use the short exact sequence of complexes of sheaves: $$\xymatrix@R=17pt { \left( 0 \rightarrow 0 \right) \ar[d] \\
\left( 0 \rightarrow \Omega ^1 _C \right) \ar[d] \\
\left( f^* \Omega ^1 _X \rightarrow \Omega ^1 _C \right) \ar[d] \\
\left( f^* \Omega ^1 _X \rightarrow 0 \ar[d] \right) \\
\left( 0 \rightarrow 0 \right) }$$

Applying the functor ${\rm Hom } ( - , {\mathcal O}_C)$ and using the long exact hypercohomology sequence we obtain: $$\xymatrix @R=3pt {& 0 \ar[r] & 
{\rm Hom } \bigl( L _f ^\bullet , {\mathcal O}_C \bigr) \ar[r] & 
{\rm Hom } (0 \rightarrow \Omega ^1 _C , {\mathcal O}_C) \\ \ar[r] &
{\rm {\mathbb E}xt } ^1 (f^* \Omega ^1 _X \rightarrow 0 , {\mathcal O}_C) \ar[r] & 
{\rm {\mathbb E}xt } ^1 \bigl( L _f ^\bullet , {\mathcal O}_C \bigr) \ar[r] & 
{\rm {\mathbb E}xt } ^1 (0 \rightarrow \Omega ^1 _C , {\mathcal O}_C) \\ \ar[r] &
{\rm {\mathbb E}xt } ^2 (f^* \Omega ^1 _X \rightarrow 0 , {\mathcal O}_C) \ar[r] & 
{\rm {\mathbb E}xt } ^2 \bigl( L _f ^\bullet , {\mathcal O}_C \bigr) \ar[r] 