Relation between mass, speed and acceleration in Special Relativity

I heard it explained that Special Relativity does not actually forbid faster-than-light travel. Instead – it was argued – it is impossible to transcend the speed of light by any massive object, because, according to Special Relativity, mass increases with speed and becomes infinite at $c$, which means a body would take an infinite amount of energy to accelerate further. However, relativistic effects only apply to external observers moving relative to the observed object.

Being in a spaceship, stationary with respect to that ship, but observing its motion relative to a star, I perceive the mass of my ship as being constant (right?). How then can I explain the fact that it’s increasingly difficult to accelerate the ship further as it travels faster and faster relative to the star? Can I use Special Relativity to predict increase in energy cost of further acceleration over increase of my speed relative to the star?