So as your speeds are >0.10c you will experience an increase of mass. I have to wonder if this relates in anyway to the resistance a magnet experiences when pushing electrons through a wire. Anyways...
So you become more massive and require more energy to accelerate any further, which makes you even more massive to require even more required energy... adding up to the point which you can no longer accelerate any further. Suppose you did though, what would happen? at v=1c it would seem you are as massive as the entire universe because your vessel is infinitely massive? But long before then it would seem your vessel would collapse on its own gravity, forming a smallest version of a black hole that we could imagine. Unsure whether or not it would be stable enough to survive, I guess it depends on where this happens and how much energy is available for the micro hole to consume. But even long before that would happen, your body would be destroyed by the massive gravitation field from you a moving through the universe, long before you could experience any relativistic time dilation affects.
What can we conclude from this? The human race is probably restricted to very low speeds, probably less than 25 percent of the speed of light. Any advanced space propulsion concepts for the future, though may become available, more efficient, propellantless, or what ever comes about, will still be subject to the law of diminishing returns, and even though we may be able to go a lot faster than chemical rockets, eventually; the speeds will still be far to low to send a man to another star. It will HAVE to be a fully life sustaining generational ship with the one-way goal of seeding the human race elsewhere.
You're apparent mass is not going to be blackhole-like until you're very close to the speed of light. Besides, your own apparent mass from your reference frame is not going to be blackhole-like.
To get a blackhole of radius 1 meter, you need a mass of 6*10^26kg. Assuming you start out at 100 tons, 1E5kg, you need a Lorenz factor (gamma) of 6e21, let's round to 1e22.
gamma = 1/sqrt(1-(v/c)^2) = 1e22.
or: 1/gamma = sqrt(1-(v/c)^2) = 1e-22,
1-(v/c)^2 = 1e-11
1 - 1e-11 = (v/c)^2
or v/c = ~(1 - 5e-12)
So in other words, you have to be faster than like 99.999999999% c to have a mass energy equivalent to a blackhole. But you're not going to collapse unless you run into something big (which would appear to YOU as having a mass-energy of a black hole). But you're going to have other problems if you run into something like that..
tl;dr: You're assuming it's linear, which is wrong.
The gravitational sources in GR are generated by the stress-energy-momentum tensor, not a scalar quantity like mass. You can always Find a transform to the frame where you are stationary, and there won't be any event horizons there.