It is perhaps easiest to understand the time-like component of the 4-velocity, with respect to some orthonormal basis, in SR. Let's say we are in the rest frame ##O'## of an inertial observer, who has setup global inertial coordinates ##(t',x^{1'},x^{2'},x^{3'})## in his frame and has constructed a set of orthonormal basis vectors ##\{e_{0'},..,e_{3'}\}## for the frame, where ##(e_{\mu'})^{\alpha'} = \delta^{\alpha'}_{\mu'}## are the components of each basis vector in this frame; you can imagine ##e_{0'}## as pointing along the "time axis" of the frame setup by the observer and the ##e_{i'}## as pointing along their respective "spatial axes". Note that since the observer is obviously at rest in his own rest frame, his 3-velocity in ##O'## vanishes and his 4-velocity in this frame is simply ##\vec{u} = e_{0'}## i.e. the 4-velocity points along the "time axis" and can be written ##\vec{u} = (1,0,0,0)## with respect to the basis vectors in the frame.
We can in fact take this to be the definition of the 4-velocity of the inertial observer in ##O'##. Then, the velocity of this inertial observer as measured by an observer in any other inertial frame ##O## (who has himself set up coordinates ##(t,x^1,x^2,x^3)## and an orthonormal basis ##\{e_{0},..,e_{3}\}## for his frame) is simply given by ##u^{\alpha} = \Lambda^{\alpha}_{\beta'}(e_{0'})^{\beta'} = \Lambda^{\alpha}_{0'}##, where ##\Lambda## is the Lorentz transformation. This is nothing more than ##\vec{u} = \gamma(1,\mathbf{v})## where ##\mathbf{v}## is the 3-velocity of ##O'## with respect to ##O##. This is of course the usual result.
Now, a particle that is accelerating has no inertial frame in which it is always at rest but what we can do is, for each event on the worldline of the particle, find an inertial frame which is instantaneously co-moving with the accelerating particle and in exactly the same way as above, we define the 4-velocity of the accelerating particle at some event as the ##e_{0}## basis vector of the inertial frame that is instantaneously co-moving with the particle at that event. Now let's tie this into the definition you are probably more used to. Let ##O## be an inertial frame (with the usual coordinate setup and orthonormal basis setup) instantaneously co-moving with our particle, at a given event, and say that the particle makes an infinitesimal displacement ##\vec{d\mathbf{x}}##. In ##O## this is written in components as ##\vec{d\mathbf{x}} = (dt,0,0,0)##. We also have that in ##O##, the infinitesimal proper time ##d\tau## is simply ##d\tau = dt## so ##\frac{\mathrm{d} \vec{\mathbf{x}}}{\mathrm{d} \tau} = (1,0,0,0) = e_{0}## which by definition is the 4-velocity of the accelerating particle at the given event.
All in all, I guess the important point to take home is that in the rest frame of an inertial observer, or in the instantaneously co-moving frame of an accelerating observer at a given event, the 4-velocity of the observer in that frame simply points along the "time axis". This is one way to interpret the time-like component of the 4-velocity.
