As the energy supply shifts from fossil fuels to inherently intermittent renewable energy sources, such as wind and solar, improvements in energy storage are required. Electrical energy storage in the form of rechargeable batteries can benefit greatly from a transition to energy dense metallic anodes that require safe, reversible, electrodeposition over many cycles of charge and discharge. In all currently used electrolytes, electrodeposition is subject to three types of instabilities: chemical, morphological, and hydrodynamic at both low and high current densities, which lead to complex transport phenomena in the electrolyte and unstable deposition, including formation of ramified structures known as dendrites. We focus on understanding the hydrodynamic instability termed electroconvection. We have developed viscoelastic liquid electrolytes to stabilize electroconvective instabilities near the electrode and improve deposit morphology. These designs consist of high molecular weight polymers that are entangled in common battery electrolytes. A key aspect of these electrolytes is the decoupling of ion and momentum transport, in which the viscosity is significantly enhanced while maintaining high ionic conductivities. These electrolytes have been studied with a metallic electrode that is subject to morphological changes, and at a more fundamental level, using ion selective membranes to isolate the hydrodynamic instability. Through these techniques the convection was found to be arrested when the polymer is present above its entanglement transition. This was further confirmed through in-situ visualization coupled with particle tracking in which the polymer significantly stabilized the motion near the ion-selective interface. The visualization was expanded to understand the convective flow structure under a variety of applied potentials and electrolyte concentrations. Through our particle tracking it has been found that electroconvection occurs on multiple length scales simultaneously. It was discovered that increases in applied potential both enhance the velocity and complexity of the convective motions. Beyond this, decreases in concentration, accompanied by larger Debye screening lengths, enhance chaos in the flow and impact the ratio of small to large convective motions in the electrolyte.