How to construct a LTI system with a specific impulse response function, say \$h(t) = \delta '(t)\$?
First, you can construct possible systems.  That means that getting an impulse response of \$h\left(t\right) = \delta'\left(t\right)\$ is right out, because it's non-causal.  For that matter, a system that truly has impulse response \$h\left(t\right) = \delta\left(t\right)\$ is also out, because anything real responds in a finite amount of time.

In general, if you can start with your desired \$h(t)\$ and break it into a weighted sum of impulse responses that you know how to realize in circuitry, then you can generally move forward.  Things of the form \$h(t) = t^n e^{at}\$ are particularly easy, because (as long as you allow \$a\$ to be complex) that's the form that can be realized by an appropriate lumped-element circuit.

You can, in theory, implement approximations of arbitrary \$h(t)\$ by making tapped delay lines and summing their outputs -- but that's kind of an expensive wacky old radar technology, not something that's of much use today (unless it's in really expensive, fast radars).

In practice, though, you rarely start with an impulse response and design a system to it.  The only time I can think of where that's really necessary is in communications systems, and these days that's generally done by converting the signal to digital and doing all of the heavy lifting with DSP.