We investigate the demand private coded caching problem, which is an $(N,K)$ coded caching problem with $N$ files, $K$ users, each equipped with a cache of size $M$, and an additional privacy constraint on user demands. We first present a new virtual-user-based achievable scheme for arbitrary number of users and files. Then, for the case of 2 files and arbitrary number of users, we derive some new converse bounds. As a result, we obtain the exact memory-rate tradeoff of the demand private coded caching problem for 2 files and 3 users. As for the case of 2 files and arbitrary number of users, the exact memory-rate tradeoff is characterized for $M\in [0,\frac{2}{K}] \cup [\frac{2(K-1)}{K+1},2]$.