The grism spectrometer works on the same basic principles as any standard grating spectrometer, so you should be able to find more information by looking at those. The main differences, of course, are that the grism spectrometer uses refractive optics and a different dispersive element.
Choosing optics and a slit for this design will involve a series of tradeoffs. It basically all comes down to throughput and resolution (and practicality, including cost). Since you're building this yourself and want to use single lenses, I'm assuming that you're sensitive to cost.
First, what spectral resolution and overall bandwidth do you need? This will determine how wide your slit can be, for a given grating (resolution is nearly proportional to slit width, assuming your camera has a small pixel size). The higher the groove density of the grating/grism, the more diffractive. So you can get the same spectral resolution with a higher groove density and a larger slit width. But the trade-off with this configuration is that you'll have a lower field of view (lower overall spectral range) since the with more diffraction you're essentially "zooming in" in the spectrum. Also, more diffractive gratings are more polarization- and frequency-sensitive in their diffraction efficiency, which may or may not be problematic.
On the other hand, if you use a narrower slit and less diffractive grating, then for the same resolution you have smaller light throughput. So, the next question would be how much light do you have? The throughput you need is generally tough to gauge since it depends on many factors. Better to just try it, but be aware of the general tradeoffs. Actually, the slit width/throughput trade-off also depends on the focusing power of your initial lens. You'll basically want to choose the focal lengths for the first two lenses (those around the slit) to focus the light enough to get most of it through the slit, and then recollimate it expanded to be about the size of the grism.
Regarding chromatic aberration, again this depends on the specifics of your setup: how much resolution you need, your bandwidth of interest, the geometry of your setup (chromatic aberration is irrelevant for a big setup with long-focal-length lenses). Decent AR-coated achromatic doublets are available for less than $100 from, say, ThorLabs, if you think it will be necessary.
You also want to make sure that spherical aberration isn't a problem. The key to this is to either use aspherical lenses or large lenses (for which the beam is a small-ish fraction of the diameter). If you have a lot of spherical aberration, the edges of your spaces image will be bad.
Good luck! Have fun!