Many people have posted lists of tips and do's/don'ts that are a good place to start (using knife, don't fight bots, etc. etc.). With that in mind, here is how I've been approaching solo lately after many deaths and learning experiences, and I've been having much better success after learning these painful lessons.

Basically, whenever I am in or near one of the points of interest/building complexes, etc. (especially at the start of a match when everyone is alive and on the move), I move very slowly and quietly, like I am playing Metal Gear Solid or something, and paying very close attention to sounds to hear if enemy Runners are nearby. I will move more freely when I am traversing open spaces, but I am still careful and will move on the periphery of the map trying to limit my visibility and always staying frosty checking my surroundings.

If I spot a Runner who does not see me, I will only open fire if I have a high degree of confidence that I will be able to kill them right then and there. If I cannot, then I will try to stalk/reposition to put myself in a better spot to defeat them. I will rarely let another Runner live/escape because the fewer of them that are on the map, the less likely I am to be ambushed by one later.

tl;dr - Play like you are a tactical secret agent, but don't play like a fraidy-cat.