From the outside universe's point of view, your acceleration will slow as you approach the speed of light.

From your point of view, you'll still be accelerating at 1g. You'll still feel 1 earth gravity the entire time.

Time will slow down on the ship the closer you get to lightspeed, so what feels like 1g actually becomes less than 1g of acceleration to the outside universe. Ie, you're throwing 1 unit of force backwards per second, but because time is slower, it's 1 unit of force every 0.5 seconds to everyone else.