Single crystals of the Ni-doped FeAs-based superconductor SrFe2-xNixAs2 were grown using a self-flux solution method and characterized via x-ray measurements and low temperature transport, magnetization, and specific heat studies. A doping phase diagram has been established where the antiferromagnetic order associated with the magnetostructural transition of the parent compound SrFe2As2 is gradually suppressed with increasing Ni concentration, giving way to bulk-phase superconductivity with a maximum transition temperature of 9.8 K. The superconducting phase exists through a finite range of Ni concentrations centered at x=0.15, with full diamagnetic screening observed over a narrow range of x coinciding with a sharpening of the superconducting transition and an absence of magnetic order. An enhancement of bulk superconducting transition temperatures of up to 20% was found to occur upon high-temperature annealing of samples.