Abstract : We study the geodesic flow of a Hilbert geometry defined by a strictly convex open set with $C^1$ boundary. We get interested in its local behaviour around one specific orbit as well as its global properties on a quotient manifold. We explain why this flow has hyperbolic-like properties, by studying in particular its Lyapunov exponents, which are linked in a precise way to the shape of the boundary of the convex. We prove an entropy rigidity result for compact quotients. We also develop general tools that can be used when considering noncompact ones, following ideas and results of negative curvature. The case of geometrically finite surfaces is studied in details, and the entropy rigidity theorem is extended to finite volume surfaces.