  Sr$_2$IrO$_4$ is an extensively studied spin-orbit coupling induced insulator
with antiferromagnetic ground state. The delicate balance between competing
energy scales plays crucial role for its low temperature phase, and the route
of chemical substitution has often been used to tune these different energy
scales. Here, we report an evolution of structural, magnetic and electronic
properties in doped Sr$_2$Ir$_{1-x}$Cu$_x$O$_4$ ($x$ $\leq$ 0.2). The
substitution of Cu$^{2+}$ (3$d^9$) for Ir$^{4+}$ (5$d^5$) acts for electron
doping, though it tunes the related parameters such as, spin-orbit coupling,
electron correlation and Ir charge state. Moreover, both Ir$^{4+}$ and
Cu$^{2+}$ has single unpaired spin though it occupies different $d$-orbitals.
With Cu substitution, system retains its original structural symmetry but the
structural parameters show systematic changes. X-ray photoemission spectroscopy
measurements show Ir$^{4+}$ equivalently converts to Ir$^{5+}$ and a
significant enhancement in the density of states has been observed at the Fermi
level due to the contribution from the Cu 3$d$ orbitals, which supports the
observed decrease in the resistivity with Cu substitution. While the long-range
magnetic ordering is much weakened and the highest doped sample shows almost
paramagnetic-like behavior the overall system remains insulator. Analysis of
resistivity data shows mode of charge conduction in whole series follows
2-dimensional variable-range-hopping model but the range of validity varies
with temperature. Whole series of samples exhibit negative magnetoresistance at
low temperature which is considered to be a signature of weak localization
effect in spin-orbit coupled system, and its evolution with Cu appears to
follow the variation of resistivity with $x$.
