A lot depends on what you mean by the term "gravitational field".
Your question has been discussed previously - see for instance
https://www.physicsforums.com/showthread.php?t=87991
You might also want to check out, for background, the simpler question of what the electric field of a moving charge looks like.
See for instance http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_15.pdf
and if you want more detail on this related topic, try looking at different page "numbers" than 15.
The short answer for a moving charge is that the parallel component of the electric field is not affected by motion, while the transverse components are boosted. One also has additional terms which are known as the "magnetic field".
In Newtonian physics, the term "gravitational field" is usually implied to mean a force. Unfortunately, there is no way to directly measure this "force" unless one has a gravitationally neutral test particle as a reference. And there isn't any such thing as a gravitationally neutral reference.
Just as GR says that it is energy, not mass, which causes gravity, GR also says that gravity is not a force, but a curvature of space-time. GR also offers a recipe to convert this curvature back into forces for slowly moving objects. Unfortunately, this recipe fails for quickly moving objects - its only an approximation.
Under most circumstances, this curvature, mathematically described by the Riemann tensor, can by physically interpreted as a "tidal force". (This identification is actually still not quite perfect, but it's reasonably close. The identification works perfectly only when the observer measuring the tidal force is free-falling).
So, while there turns out not to be any unambiguous way to determine the "force" of gravity of a moving object (I have a rather long-running argument with another poster on this point, BTW), it is quite possible (and in the context of GR, even natural) to talk about the "gravitational field" in terms of the Riemann tensor, or in terms of tidal forces.
A tidal force is just the force/unit length caused by the change in gravity. For instance, if you stand in the Earth's gravity field, your head is further away from the center of the planet than your feet. This causes a slight, but measurable difference in an accelerometer mounted at your feet and and a different one mounted on your head. This difference, per unit length, is just a tidal stretching force. While it is very small in magnitude, it can be measured with a sensitive enough instrument (such as a Forward mass detector).
Now that we've discussed the background, we can talk about the actual result of the gravitational tidal force generated by a moving mass. The result is not intuitive - it says that if you directly approach or move away from a massive object, there is no effect on the tidal force(s) at all.
If you move tangentially to the object, the tidal force(s) increases.
Thus, for instance, an observer falling directly towards a black hole (following a geodesic) at nearly light-speed will not see any change in the tidal force from that black hole. (This is in the textbooks, MTW's "Gravitation" discusses this).
However, an observer moving in such a manner as to orbit a black hole at nearly light speed (still following a geodesic) will see an increased tidal force from the black hole. (This example is not in the textbooks, but can be worked out with enough knowledge and effort).
