An object moving with constant proper acceleration will never reach c. The math is worked out [here](http://aether.lbl.gov/www/classes/p139/homework/eight.pdf). Assuming it starts from rest, it will move along a hyperbola, and its speed will asymptotically approach c.
Near, but never quite to c. Specifically, the ship's velocity at constant acceleration can be defined as v=c*tanh(a*t/c). After 2 years at 1g acceleration your ship will be at nearly 97% of the speed of light. More on constant acceleration spacetravel here: https://en.wikipedia.org/wiki/Space_travel_using_constant_acceleration