We investigate the ability of the Laser Interferometer Space Antenna (LISA) to detect a stochastic gravitational-wave background (GWB) produced by cosmic strings, and to subsequently estimate the string tension $G\mu$ in the presence of instrument noise, an astrophysical background from compact binaries, and the galactic foreground from white dwarf binaries. Fisher Information and Markov Chain Monte Carlo methods provide estimates of the LISA noise and the parameters for the different signal sources. We demonstrate the importance of including the galactic foreground as well as the astrophysical background for LISA to detect a cosmic string produced GWB and estimate the string tension. Considering the expected astrophysical background and a galactic foreground, a cosmic string tension in the $G\mu \approx {10}^{-16}$ to $G\mu \approx {10}^{-15}$ range or bigger could be measured by LISA, with the galactic foreground affecting this limit more than the astrophysical background. The parameter estimation methods presented here can be applied to other cosmological backgrounds in the LISA observation band.