From initial rate data, comparing experiment 1 and 2, it is seen that rate doubles as [CH3I] doubles. Comparing experiments 2 and 3, it is seen that rate doubles as [DNA] doubles. These data show that the rate is first order in both reactants, and the rate equation can be written as rate = k[DNA][CH3I].
Mechanism I would be consistent with the rate law, but not with the concept that CH3I interacts with DNA, because the DNA appears as a product in mechanism I indicating no change in the DNA. Excluding the biological implications, and only looking at the kinetics of the reaction, Mechanism I would be a possible mechanism.
Mechanism II is not consistent with the initial rate data. The rate equation for the elementary rate limiting step in this proposed mechanism is rate = k[CHI] which is NOT the same as the actual rate equation derived from the data. Furthermore, DNA does not appear in the final reaction according to mechanism II.