Modern particle accelerators are a key component of today’s research landscape and indispensable in industry and medicine. In special application areas, the portfolio of these facilities will be expanded by laser-driven compact plasma accelerators that generate short, high-intensity pulses of ions with unique beam properties. Though intensely explored by the community, scaling the maximum beam energies of laser-driven ion accelerators to the required level is one of the most significant challenges of this field. This endeavor is inherently linked to a fundamental understanding of the underlying acceleration processes. The prospect to efficiently increase the beam energy relies on the ability to control the accelerating field structures beyond the well-established acceleration from the stationary target rear side. However, manipulating the interaction in such micrometer-sized accelerators proves to be challenging due to the transient nature of the plasma fields and requires precise tuning of the temporal laser pulse shape and the volumetric density distribution of the plasma target to a level that could so far not be achieved. This thesis investigates laser-proton acceleration using a cryogenic hydrogen target that combines the capabilities of predictive three-dimensional simulation and the in-situ realtime monitoring of the density distribution in the experiment to explore the fundamental physical principles of plasma based acceleration mechanisms. The corresponding experiments were performed at the DRACO laser facility at the Helmholtz-Zentrum Dresden-Rossendorf. The key to the success of these studies was the advancement of the cryogenic target system that generates a self-replenishing pure hydrogen jet. Using a mechanical chopping device, which protects the target system from the disruptive influence originating from the high-intensity interaction, allowed, for the first time, systematic experiments with a large number of laser shots in the harsh environment of the ultra-short pulse DRACO petawatt laser. The performance of a cylindrical hydrogen jet can be substantially optimized by a flexible all-optical tailoring of the target profile. Guided by real-time multi-color probing, the target density, the decisive parameter of the interaction, was scanned over two orders of magnitude allowing the exploration of different advanced acceleration regimes in a controlled manner. This approach led to the experimental realization of proton beams with energies up to 80 MeV and application relevant high particle yield from advanced acceleration mechanisms occurring in near-critical density plasmas, a regime so far mostly investigated in numerical studies. Besides cylindrical jets, the formation of thin hydrogen sheets was studied to gain insight into the fluid and crystallization dynamics that can be used to tailor the target shape for laser-proton acceleration. Using these jets, the onset of target transparency was explored, a regime that promises increased proton energies when optimized. Furthermore, after irradiation of the hydrogen jet with a high-intensity laser pulse, an unexpected axial modulation in the plasma density distribution was observed that can play a role in structuring the proton beam profile. This modulation is caused by instabilities that originate from the laser-plasma interaction, for example due to laser-driven return currents or the plasma expansion dynamics.