Highest Trope-ic Level

"trophic level - noun - ecology - Each of several hierarchical levels in an ecosystem, consisting of organisms sharing the same function in the food chain and the same nutritional relationship to the primary sources of energy."

Apex; writhing, tentaculate, and for the longest time, vexing for me to try and figure out by just looking at its cards on paper, and in a vacuum.

Now, after playing a fair few games with it, I see its strength.

Apex is pure adaptability; able to gather strength to be able to puncture the defences of strong glaciers yet also capable of keeping pace with fast advance. All the while possessing tankiness that is hard to simply flatline.

Geist: "What the hell happened? Toro flatlined. You were meant to just case the joint".
Anna: [incoherent rambling]
Jorge: "She says the net... it just came alive and took him."

Level 1: Autotrophy - Making your own food

Apex is strong: but like any runner it needs to set up its own base of resources before being able to start its assault.

Priorities for starting the game are:
1. Sure Gamble
2. ♦ Heartbeat
3. Data Folding, Hunting Grounds or Wasteland
4. Trope
5. Harbinger or Leprechaun

Anything else is, at this stage, facedown fodder. Your main priorities at the moment are installing as many face-downs as possible (use Endless Hunger face-down if you have to). The only exceptions to this are your 2/1-ofs and maybe a single copy of Apocalypse if you are dead set on that course of action.

Your other priority is getting your credit stream down, Data Folding will supply you with ample money for the rest of your game and once you start making runs, Wasteland will start making things ridiculous.

Once you have a few face-down installs, a Wasteland or two and a Heartbeat down, it's time to start facechecking.

Level 2: Herbivory - Eating the defenceless

Apex natively has the tools to force his way in and overpower a wide slew of ice. Traditional, simple, early-game, "↳End the run." ice might as well not be there and with the addition of e3 Feedback Implants even normally hard-taxing ice like Heimdall, or Archer become merely annoying.

Even without Endless Hunger and e3, Apex probably has the safest facechecking potential this side of Kit once you get Heartbeat up. Just be sure to have a decent spread of face-downs, and be very wary if you installed a Trope, respect destroyer ice.

When trashing face-downs, don't forget that you can look at the underside whenever you want, try to free up cards you will be wanting back with Trope in the future.

Level 3: Carnivory - Eating the Eaters

So, most common or garden variety ice are now neutered with your native suite of Endless Hunger, Hunting Grounds, Heartbeat and e3, that's great!

But guess what.. there are still bigger fish...

As mentioned before, destroyer ice can cripple Apex; to use a comprehensive suite of breakers with Endless Hunger you have to use Leprechauns or lots of hardware memory (which makes tanking Cortex Lock with Heartbeat unbelievably expensive).
This is why we have 3x Sharpshooter, and you should have at least 1 installed before you attempt any serious running.

Eliminating destroyers leaves the next scariest type of ice... AP subroutines. Wait what?
I know I just said that Apex can take a beating thanks to Heartbeat, but there are limits. No only can some ice do enough damage to overwhelm your board state. Some come bundled with very painful effects, overlapping with destroyers, or not.
Regardless you will want Deus X, even if it's redundant for some ice, you can save Sharpshooters with them.

The third clade of ice that will seriously impede you are... tracers.
I know this seems strange, but tracers are the only remaining means for the corp to kick you out of a server, and if they can get a significant enough head start on you economically (which they can and will do if given a chance) you will be defeated again and again by ensconced "end the run"s and tag punishment of various types. Not to mention this guy. Gingerbread, though unorthodox in other runners is probably the best way of dealing with them here.

Level 4: Still vulnerable

This leaves few scary ice left. Wotan is hardly played by anyone. And almost anything else can be Preyed upon after suffering them once. But there is one outlier... Turing... dear god, Turing...

Turing is played frequently enough to see multiple installs of it in a single game. And thus far we have no answer to it other than losing your entire turn. Kiss goodbye to your tempo, kiss goodbye to Apocalypse.
Two on a remote server will waste you 2 whole turns and a Prey to get into, and if there's an Architect there, you're permanently locked out.
Enter Refractor and Ghost Runner, these beauties can get you past in the same click cheaply with 2 stealth credits, not to mention they can be used to break any other code gate that you might not want to waste a disposable on. Ghost runner also gives you an injection of cash you may need before and during a run (if it is depleted), something the commonly taken Dirty Laundry can't boast.

You could go 2x for either Refractor or Gingerbread to aid reliable drawing, but please, in that case, know what your weaknesses are and play around them.

Level 5: Autophagy - A word on self consumption

I hope this guide has cleared up my deck, but I sense you have one last query...

"Why Trope?"

On the surface, it may seem like a very strange include considering Levy AR Lab Access exists and is the go-to solution for wholesale deck recursion. Something Apex needs when face-downing a card nearly every turn and using disposable breakers and trash effects.

Two reasons: precision and reliability.

First, precision. Trope's main purpose is to be used prior to an Apocalypse turn to refill your stack with the breakers, events and hardware that will be necessary to rebuild in the aftermath. Anything left face-up will be locked out from you forever if you can't get another Endless Hunger or Heartbeat. For this reason, it is best to keep those cards in hand and ready to outpace the corp in the turns immediately following. Levy AR Lab Access clumsily snatches those vital components out your hand introducing a large component of risk into the equation.

Second, reliability. Whilst Levy's strength is that it takes everything back to the stack, you often need certain cards more than others, after an Apocalypse, you don't need more face-downs, you'll be swimming in them! What you'll need are those cards needed to rebuild, nothing more. An additional point of note is that with Trope installed on the board you can free up an additional slot in your grip (remember one will be taken up with the #Apocalypse you're planning on using).

Just remember, on your apocalypse run, be sure you can pull it off. Make sure to trash the installs (especially resources) you need after. Try to have your Heartbeat ready.

Jenkins: "Ash, can you hear me? Ash?"
Ash: [speaking in an electronic, distorted voice] "Yes, I can hear you."
Jenkins: "What was your special order?"
Ash: "You read it. I thought it was clear."
Jenkins: "What was it?"
Ash: "Exclude digital life form. Priority One. All other priorities rescinded."
Keegan: "The damn company. What about our livelihoods, you son of a bitch? You nearly bankrupted us running a single trace!"
Ash: "I repeat, all other priorities are rescinded."
Jenkins: "How do we kill it Ash? There's gotta be a way of killing it. How? How do we do it?"
Ash: "You can't."
Keegan: "That's bullshit."
Ash: "You still don't understand what you're dealing with, do you? Perfect organism. Its structural perfection is matched only by its hostility."
Mai: "You admire it."
Ash: "I admire its purity. A survivor... unclouded by conscience, remorse, or delusions of morality."
Keegan: "Look, I am... I've heard enough of this, and I'm asking you to pull the plug."
Ash: [Jenkins goes to disconnect Ash, who interrupts] "Last word."
Jenkins: "What?"
Ash: "I can't lie to you about your chances, but... you have my sympathies."

Thanks to @tendermovement for originally suggesting Leprechaun and disposable breakers.