This paper is the first of a series of papers constraining cosmological parameters with weak lensing peak statistics using $sim 450~rm deg^2$ of imaging data from the Kilo Degree Survey (KiDS-450). We measure high signal-to-noise ratio (SNR: $ u$) weak lensing convergence peaks in the range of $3< u<5$, and employ theoretical models to derive expected values. These models are validated using a suite of simulations. We take into account two major systematic effects, the boost factor and the effect of baryons on the mass-concentration relation of dark matter haloes. In addition, we investigate the impacts of other potential astrophysical systematics including the projection effects of large scale structures, intrinsic galaxy alignments, as well as residual measurement uncertainties in the shear and redshift calibration. Assuming a flat $Lambda$CDM model, we find constraints for $S_{rm 8}=sigma_{rm 8}(Omega_{rm m}/0.3)^{0.5}=0.746^{+0.046}_{-0.107}$ according to the degeneracy direction of the cosmic shear analysis and $Sigma_{rm 8}=sigma_{rm 8}(Omega_{rm m}/0.3)^{0.38}=0.696^{+0.048}_{-0.050}$ based on the derived degeneracy direction of our high-SNR peak statistics. The difference between the power index of $S_{rm 8}$ and in $Sigma_{rm 8}$ indicates that combining the two probes has the potential to break the degeneracy in $sigma_{rm 8}$ and $Omega_{rm m}$. Our results are consistent with the cosmic shear tomographic correlation analysis of the same dataset and $sim 2sigma$ lower than the Planck 2016 results.
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