Line Bundle on subvarieties
Solution 1:
In general (i.e. for restricting line bundles from a variety to a subvariety) one has $c_1(L_{\vert X}) = c_1(L) \cdot X$ (the intersection, thought of as a divisor on $X$).
In your case one can be more explicit, since $L = \mathcal O(d) = \mathcal O(1)^{\otimes d}$ for some $d$ (all line bundles on projective space are of this form). Thus, if we let $H_X$ denote a generic hyperplane section of $X$, then $c_1(L_{\vert X}) = d H_X$.