Poor Exile. Poor, poor Exile. On the face of it, his ability is actually pretty decent--click compression is always nice, and drawing cards is nothing to sneeze at. Plus, installing programs from the heap is a powerful Shaper tool that you'll want to make use of anyways--just look at Clone Chip!
That is, unfortunately, where Exile's problems start. Installing programs from the heap is strong. Which means that there's not actually that many ways to do it: in-faction you have Clone Chip, Test Run, and Exile's own signature card, Scavenge. They're all good, but because that's all there is, your ability reads: "Over the course of the game, draw maybe nine cards". Still nice, but that's just not enough for you to want to pick him over any other Shaper. So we're going to have to go deeper.
Out of faction, we find an astounding two cards that install programs from the heap: Retrieval Run, which is two influence and really awkwards besides, and Pawn, which only recurs other Caïssa programs--but that includes other Pawns. What you can do, then, with Deep Red as your console, is continuously loop Pawns. If they have a one-ICE central sitting around, every run you make nets you a free draw, a credit off of Scheherazade, and another counter on all of your Technical Writers. That's pretty darn good! But boy, that's a lot of red cards...
Which unfortunately brings us to our next problem: Exile has no influence. Sure, it may say he's got the standard 15, but the Clone Chips are absolutely mandatory. And then you're importing all of the pieces of the Street Chess engine, which given that you probably want three Deep Red and a full suite of Pawns, and a single Scheherazade, is another 7 influence. So now you've got all of 5 influence to play with--and one of those pips is likely going to Lady. Still, 4 pips to play with isn't nothing, right?
Except that now we're running into another problem: actually taking advantage of all of our heap installs. Refreshing Lady is a nice trick, but we can't really go full Dog Mode with Cuj.0 and Rex--after all, we don't have the influence. Bouncing a Cyber-Cypher with a Scavenge is a good trick, but what we really need is some way to trash programs for value--well, Aesop runs a Pawnshop for a reason, right? And Cache is a card you'll happily recur again and again, and then sell off! Excellent! Now we just need to import a card or two for our breaker suite, and... sorry, what's that you say? We only have one influence left? And we still don't have a sentry breaker? Uh-oh. Could this be Creeper's time to shine? (No. No, it is really kind of not--8 credits to break an Ichi 1.0, oh boy!). Those four points from the NAPD are really hurting right now.
Of course, you could abandon any part of this engine, but that just brings you back around to your first problem: why are you playing Exile in the first place? You've had to jump through about twenty hoops, and now what you've got to show for it is an admittedly pretty strong engine, a point of link, and a breaker suite that you only had about two free influence for. Now, the deck is quite fun, don't get me wrong. It can and does win games! But at the end of the day, you're playing Exile because you want to play Exile (maybe out of love for that collared shirt plus hoodie combo), not because Exile is a competitive option. But hey, even if he's garbage-tier, at least he's in his element, right?