The most obvious thing is consistent levels. You don't want jumps in volume, and you don't want plosives.
Start with the recording. You need a good pop filter, and you want someone with the right kind of voice who knows how not to pop their plosives too much.
Then a large-diaphragm microphone is your friend, ideally a dynamic mic like an SM7 (the classic radio mic). There's no reason you can't make do with any other kind of mic, but a larger diaphragm mic tends to roll off the higher frequencies in a smooth, natural way, and a dynamic mic does that even more. With a cardioid mic you also get proximity effect which lifts the bass and low-mids, which can help the "depth" of the voice. Too much makes it boomy though. If you want to stop them doing this, put the pop filter where you want their mouth to be.
Then look at EQ. Most people don't have anything going on below 100Hz vocally, so a high-pass there is good to get rid of ambient noise in the recording from aircon, footfall, and so on. If the low-mids are sounding muddy, you might want to roll out a bit (although ideally just get the talent to move away from the mic a bit). You might want to roll out a bit gently around 3-5kHz (the "baby crying" resonant frequency of our ears) to make it less harsh, although if you take out anything significant then you're probably losing clarity on the voice so be cautious. And a gentle roll-off above 10kHz will smooth out the edges and reduce recording hiss. Beware with all of this that things still need to be clear and audible though.
Then look at compression. This will even out jumps in volume. A two-stage process is good here: one stage to react quickly to transients above some level, maybe even a limiter; and a slower stage to even out the general volume. On speech, if the talent knows what they're doing then you'll likely only be looking at about 3dB on each compressor.
And then decide where you want this spatially. Do you want it ultra-dry, like they're whispering into your ear? Do you want a touch of short, well-damped reverb for a bedroom? Or do you want to add some low wind and nature sounds, with a subtle well-damped echo to suggest trees around you?
Mainly though you need to start with the voice. With a good voice, all you need to do is add some sparkle and try not to kill it. If you haven't got that, you can polish the turd as much as you like, but all you've got at the end is a sparkly turd. :)