The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to read this popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".
What actually changes at relativistic speeds is the dynamical law that relates momentum and energy depend with the velocity (which was already written). Let me put it this way: trying to ascribe the modification of the dynamical law to a changing mass is the same as trying to explain non-Euclidean geometry by redefining $\pi$!
Why this law changes is the correct question, and it is discussed in the answers here.
So when people say: 'I am approaching the speed of light, and to get to 100% light I would need infinite energy' they are essentially saying that this situation is impossible?
Yes.
I read this in Hawking's book and confused me because I assume when he says 99.9% speed of light, he means 99.9% speed of light in relation to someone outside observing?
Yes, but note that $c$ is a universal constant. If something is traveling at the speed of light, it is traveling at the speed of light to everyone (except other photons traveling parallel, see my answer to this question).
I just cannot understand this notion of needing more and more energy to get closer to light as absolute velocity does not exist? (in that it is a purely relative concept).
It boils to relativity. The energy of a particle is related to the velocity, $v$, via the relation
$$
E=\gamma mc^2=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}
$$
As $v\to c$, $E\to\infty$, but gives an indeterminate operation at $v=c$.
Surely the ability to accelerate further cannot possible be impeded because speed is all relative, there should be no limit to acceleration? If I 'accelerate' a further 50MPH, will I get to the destination exactly 50 miles early?
(a) 50 mph is a speed, not an acceleration and (b) you can't get somewhere "50 miles early." If your trip is 300 miles, you can't get there in 250 miles. You can get there in a shorter amount of time.
From what I can gather you 'can' accelerate FTL (sort off) but instead space bends towards you so you will get to your destination 'ftl' but only due to the curvature in space? So in effect, you can go light years in seconds (lets forget the practicals for a second), but from anyone observing, this will ALWAYS take light years.
The 2nd postulate of special relativity states that $c$ is invariant of reference frame (constant for everyone), so nothing can accelerate to speeds faster than light.
Also, if for me I am going 'FTL', does outside observers see me as going light speed, or is it 99.999%.. is there a specific number?
Faster than light means that you have a speed $v>c=2.9979\times10^{10}$ cm/s. If you have done this, then kinematics suggest you have a complex velocity (as in the imaginary number complex) which is utter nonsense since velocity is a physical (real) quantity.
Best Answer
The answers given so far are fine, but to my surprise nobody's mentioned the most important point: in modern terminology, we generally don't say that the mass of an object increases with speed. "Relativistic mass increase" is outdated terminology, not used by most physicists anymore. In general, nowadays, "mass" means "rest mass" and is independent of velocity. Igor Ivanov's answer to this question says it all.
I haven't read the article by Lev Okun that he refers to, but I like the term "pedagogical virus" for this notion.