Measurements of 119Sn nuclear spin-lattice relaxation rate, 1/T1, and Knight shift, K, in the offstoichiometric compound CeNi1.01Sn and the substituted compounds CeNi1-xMxSn (M=Cu,Co) and Ce1-xLaxNiSn have been made in order to unravel the impurity and/or carrier doping effects on the V-shaped gapped state in CeNiSn. From the detailed analysis of the T dependence of 1/T1, it is shown that the density of states (DOS) is induced just at the Fermi level for CeNi1.01Sn, whereas the DOS increases progressively with the dopant in a finite-energy range near the Fermi level for the substituted compounds. The respective substitution of Co and La into Ni and Ce sites changes the gapped state into the nonmagnetic Fermi-liquid state, whereas the replacement of Ni by Cu with x> 0.06 gives rise to the antiferromagnetic (AF) ground state. It is suggested that the AF order is realized by the combined effect of the rapid collapse of the V-shaped gap and the increase of the DOS at the Fermi level. It has been found to lead to the magnetic ground state and the nonmagnetic Fermi-liquid state to dope electrons and holes into the renormalized conduction bands, respectively.