No- if you think about a 'classical' particle (that is, ignore quantum effects), once you are inside the particle the gravitational pull is proportional to the distance from the center.
In other words, if you are outside a spherical mass the pull of gravity gets stronger as you get closer (~1/r^2). But once you are inside the mass, the pull gets weaker. Hence there is no black hole at the center of the earth, for example.
you are correct. But what about objects moving past me at near light speeds? Would they see my mass increase to the point that I have an event horizon?
No, because gravitational mass is not the same as (apparent) relativistic mass. I wrote a post on it a while back, but I'm not sure of the protocol regarding comment links.
In other words, if you are outside a spherical mass the pull of gravity gets stronger as you get closer (~1/r^2). But once you are inside the mass, the pull gets weaker. Hence there is no black hole at the center of the earth, for example.