Precise Point Positioning (PPP) is a method that performs precise position determination in static or kinematic mode using a single GPS receiver. It has been recognized as a very valuable tool in many applications such as in geodetic and geodynamic, in geophysics and meteorology. This paper focuses on a sequential least squares estimation for unknown parameters and accuracy analysis of GPS kinematic Precise Point Positioning (PPP) with marine data. A least squares solution with a priori weighted constraints to the parameters is mainly introduced. The adjustment procedure which is an effectively sequential filter, considers the variations in the states of the parameters between observation epochs and uses appropriate stochastic process noise to update their variances. Combining IGS precise orbit and clock products with ship-borne GPS dual-frequency code and carrier phase observations collected in the sailing and floating state, the validity of the sequential least squares solution and accuracy of GPS kinematic PPP is tested and verified. The results show that the precision of centimeter can be obtained, which can meet the requirements of oceanographic kinematic users.