Abstract : In this thesis, we study some aspects of local analysis in almost complex manifolds. We first study the cotangent bundle which is a fundamental tool for complex analysis and geometry. We construct a lifted almost complex structure, using a connection on the base manifold; this unifies the complete lift defined by I.Sato and the horizontal lift introduced by S.Ishihara and K.Yano. Moreover, we study some geometric properties of this lift and its compatibility with symplectic forms on the cotangent bundle. In the next chapters, we are interested in local analysis of pseudoconvex domains of finite D'Angelo type in a four dimensional almost complex manifold. We construct local peak plurisubharmonic functions, generalizing a result of J.E.Fornaess and N.Sibony. Such plurisubharmonic functions give attraction and localization properties for pseudoholomorphic discs. In particular, this reduces the study of the Kobayashi pseudometric to a purely local problem. The Kobayashi pseudometric is an important tool for the study of pseudoholomorphic maps and for the classification of domains, and gives informations on the geometric and dynamic properties of the manifold. We give local estimates of this pseudometric on a neighborhood of the boundary, and, for a strictly pseudoconvex domain, we obtain sharp estimates. As an application we study the links between the Kobayashi hyperbolicity and the Gromov hyperbolicity; we generalize, in the almost complex setting, a result of Z.M.Balogh and M.Bonk.