how would severe sleep deprivation manifest? Like, if a character was continuously woken up every time they tried to sleep, or given a chemical that makes it so that they can't fall asleep?
It depends on how long they’re kept awake.
Missing a single night of sleep can result in mood swings and a marked drop in alertness. On a physical scale, you get muscle cramps, headaches, lightheadedness, and nausea. 24 hours of wakefulness can lead to cognitive impairments on par with being drunk.
After a few days, the fun really starts. You can begin to experience “microsleeps”, which are short periods of sudden sleep lasting just a few seconds at a time. During these, you can remain sitting or standing, but your awareness blinks out.
Your oxygen intake is also lessened, which limits your athletic potential and coordination.
Prolonged sleep deprivation can even cause symptoms of psychosis, hallucinations, paranoia, etc.