The explanation of the accelerated expansion of the Universe poses one of the most fundamental questions in physics and cosmology today. If the acceleration is driven by some form of dark energy (DE), and in the absence of a well-based theory to interpret the observations, one can try to constrain the parameters describing the kinematical state of the universe using a cosmographic approach, which is fundamental in that it requires only a minimal set of assumptions, namely to specify the metric, and it does not rely on the dynamical equations for gravity. Our high-redshift analysis allows us to put constraints on the cosmographic expansion up to the fifth order. It is based on the Union2 type Ia Supernovae (SNIa) data set, the Hubble diagram constructed from some gamma ray burst luminosity distance indicators, and Gaussian priors on the distance from the baryon acoustic oscillations, and the Hubble constant h (these priors have been included in order to help break the degeneracies among model parameters). To perform our statistical analysis and to explore the probability distributions of the cosmographic parameters, we use the Markov Chain Monte Carlo method (MCMC). We finally investigate implications of our results for the DE; in particular, we focus on the parametrization of the DE equation of state (EOS). Actually, a possibility of investigating the nature of DE lies in measuring the DE EOS, w, and its time (or redshift) dependence at high accuracy. However, since w(z) is not directly accessible to measurement, reconstruction methods are needed to extract it reliably from observations. Here we investigate different models of DE, described through several parametrizations of the EOS, by comparing the cosmographic and the EOS series. The main results are as follows: (a) even if relying on a mathematical approximate assumption such as the scale factor series expansion in terms of time, cosmography can be extremely useful in assessing dynamical properties of the Universe; (b) the deceleration parameter clearly confirms the present acceleration phase; (c) the MCMC method provides stronger constraints for parameter estimation, in particular for higher order cosmographic parameters (the jerk and the snap), with respect to those presented in the literature; (d) both the estimation of the jerk and the DE parameters reflect the possibility of a deviation from the ΛCDM cosmological model; (e) there are indications that the DE EOS is evolving for all the parametrizations that we considered; (f) the q(z) reconstruction provided by our cosmographic analysis allows for a transient acceleration.