For any smooth fiber bundle
$$ F\hookrightarrow P \stackrel{\pi}{\to} M $$
we have a short exact sequence of vector bundles over $P$
$$ 0\to VTP\to TP \to \pi^* TM\to 0, $$
where $VTP$ denotes the vertical tangent bundle defined as the kernel of the differential of $\pi$. If the bundle is holomorphic then the above is a short exact sequence of complex vector bundles and we deduce
$$ c(TP)= c(VTP)\cdot \pi^* c(TM). $$
The classical Euler exact sequence argument shows that when $P=\mathbb{P}(V)$ that $\newcommand{\bC}{\mathbb{C}}$
$$ \gamma^*\otimes \pi^*V \cong \underline{\bC}\oplus VTP, $$
where $\underline{\bC}$ denotes the trivial line bundle. Hence
$$ c(TP)= c(\gamma^*\otimes \pi^*V)\cdot \pi^* c(TP). $$
In Section I.3 of Fulton-Lang Riemann-Roch algebra you can find an explicit formula for $c_k(L\otimes E)$, $L$ line bundle and $E$ vector bundle of rank $m$. More precisely
$$ c_k(L\otimes E)=\sum_{j=1}^k \binom{m-j}{k-j} c_j(E)c_1(L)^{k-j}. $$ Note. The original answer had an error that I have now corrected. (Hat tip to abx).