We report on X-ray powder diffraction, magnetic susceptibility, electrical resistivity, and specific heat measurements on samples of Ce1-xLaxPt3Si and CePt3Si1+δ in order to better understand this unusual heavy Fermion superconductor’s normal and superconducting state properties. By suppressing the antiferromagnetic transition, TN = 2.2 K, using La-doping on the Ce-site in CePt3Si, we find that in Ce1-xLaxPt3Si, x ≥ 0.8, ρ, χ, and C/T show non-Fermi liquid temperature dependences. C/T ~ Tα with α ~ 1 for the temperature range of 0.3–3 K while low-temperature (2–20 K) susceptibility data follow 1/χ−1/χ0 = aTη behavior with η = 0.36. In addition, χ and C/T exhibit single-ion behavior in this composition range (x ≥ 0.8), i.e. the measured values expressed per Ce-mole are independent of x. Via doping the Si-site with a small excess—2–4—of Si, we present specific heat data for the bulk superconducting transition at Tc ~ 0.8 K showing that this excess sharpens the transition, obviating the need for annealing. Measurements of the X-ray diffraction patterns, ρ, χ, and C/T of these CePt3Si1+δ samples, when compared to similar measurements on annealed CePt3Si samples, indicate that the microscopic mechanism for the strengthening and sharpening of ΔC(Tc) with Si-excess may be similar to that responsible in the annealed material.