Maybe the thing you are seeking for is a dynamics of a tachyon. I will briefly introduce it. Below I use $m=\mathrm{const}$ as a rest mass or simply the mass (mass changing with speed is a pedagogical mistake), and $c=1$ - a usual convention to simplify formulas without information loss. $\gamma$ is an abbreviation for $1/\sqrt{1-v^2}$ and is called Lorentz factor (for the chosen $v$).
Mass $m$ of a particle plays that role that it fixes the relation between particle's energy and momentum: $E^2-|\vec{p}|^2=m^2$. Energy and momentum, taken together, make an energy-momentum four-vector, with an energy value counted as a 4th component (usually called 0th, and written at the beginning, in a $(E,\vec{p})$ or $(E,p_x,p_y,p_z)$ manner; graphically it is usually drawn vertical). This four-vector is important because it is tangent to the world line - a trajectory in space-time - and hence its direction gives a velocity and the speed: $v=p/E$.
A force applied to the particle deals with its energy-momentum four-vector. It tryes to change it. But the energy-momentum vector has a constraint shown above, so force cannot give it any value - only allowed ones. In the meantime, within that constraint, any value is reachable, and in many ways - with uniform acceleration, non-uniform, non-unceasing, and so on.
So we have to look closer at the energy-momenum values. For a non-zero mass, $E^2-|\vec{p}|^2=m^2$ describes the two-sheet hyperboloid (only an upper sheet is taken to get the correct energy sign). The energy-momentum vector cannot tilt lower than to slope ratio $E/p=1$ (actually it is always even higher, $E/p>1$), hence the limitation on the speed: it cannot be more than 1 (in our convention, that is the speed of light $c$). But we can consider a theoretical possibility of a particle that is different from the very beginning, $E^2-|\vec{p}|^2=\mathrm{const}<0$, that would formally give $m$ an imaginary value. Such a particle is called tachyon (though no such particles are found yet, and most physicists believe there would not be found any in the future - because of advanced reasons). It would "live" on the one-sheet hyperboloid, instead of the two-sheet one, and would never become "normal".
All the formulas for such particle stay the same, if we just assign $m$ and $\gamma$ imaginary values - and in fact imaginarities cancel in many formulas. For example, for the acceleration collinear with the velocity,
$$F=m\gamma^3\frac{dv}{dt}$$
and for the acceleration perpendicular to the velocity,
$$F=m\gamma\frac{dv}{dt}$$
(a total formula just decomposes the total force into collinear and perpendicular parts). Since $m$ and $\gamma$ are both imaginary, both $F$ and $dv/dt$ end up being real.
Besides that, tachyon shows some peculiar properties, following from its definition and formulas. It cannot move slower than light. It can move with infinite speed. It can change its direction of motion with respect to time - it can be accelerated to "faster than infinity", that would appear as a large velocity in the opposite direction, and in the preceding instants of time. In fact, an observer would see two tachions approaching each other and then disappearing. Or else, two tachions could appear from nowhere and start to move away from each other. (Remember that this picture ignores some other aspects and issues.)
Now some formulas so you can work with them yourself. SR particle mechanics is written with respect to the proper time, which given by
$$d\tau^2=dt^2-(dx^2+dy^2+dz^2)\qquad\qquad\gamma\,d\tau=dt$$
Four-force is (by definition)
$$f^\mu=(\gamma P,\gamma\vec{F})$$
where $P$ is a mechanical power of that force. Four-velocity is (by definition)
$$u^\mu=(\gamma,\gamma\vec{v})$$
We would also need a derivative
$$\dfrac{d}{d\tau}=\dfrac{dt}{d\tau}\dfrac{d}{dt}=\gamma\dfrac{d}{dt}$$
And now, the four-acceleration is (by definition)
$$a^\mu=\dfrac{du^\mu}{d\tau}$$
(you can find it in explicit form), and the Newton's second law is
$$f^\mu=\dfrac{dp^\mu}{d\tau}=m\dfrac{du^\mu}{d\tau}$$
(with $p^\mu$ meaning an energy-momentum four-vector). You can find the explicit equations for spatial values $\vec{F}$ and $\vec{a}$.
