For what it's worth, here's how I got the Lazer jet:
First, you're gonna want to build up a moderate amount of cash. $75,000 should be enough to get all the upgrades you'll need to make a good stab at this, so you're better off waiting until you've finished the first heist or collected a few hidden packages. I would also recommend getting to the point in the story where Trevor is unlocked and past the missions where you can once again switch between all three characters (not that you're gonna need Michael or Franklin, but for the sake of complete freedom).
Get Trevor's off-road Canis and take it to your nearest Los Santos Customs. I've tried infiltrating Fort Zancudo with Franklin's car and bike and Michael's car, both upgraded to the max, but what they make up for in speed, they lack in pure robustness. Trevor's canis can be fitted with a turbo and a few other engine upgrades to max out its acceleration. I would recommend upgrading the rest of it as well (it'll cost roughly $50,000) and getting 100% armour and bulletproof tires. Trust me, this will be a GODSEND in Fort Zancudo.
Once you've done that, head to your nearest Ammu-Nation and buy yourself the super-heavy bullet-resistant vest ($2,500). This is absolutely essential. You won't make it past the first barrier if you don't. If you feel you need the backup of some firepower, grab 500 rounds of your favourite rifle (depending on what you've unlocked to that point). It's not necessary, but it's always good to have a couple of thousand Micro SMG rounds at hand so you can blast any soldiers standing by the jet from the relative safety of your vehicle.
Here's where you need to make a choice. There are two entrances to Fort Zancudo: one across a rumbling bridge to the east, and the other from the furthermost western point of the map. This may vary for everyone, but what I've discovered is that if you come in from the western side, you will meet at least one tank en route to the jet's location. One shot from this and you're toast. Coming in across the bridge to the east requires more skill and expert timing, but it's a straightforward run to the jet once you've penetrated the main part of the airbase. More on that in a minute. Whatever your choice, the first thing you're gonna want to do is quick save before you enter the base. That way, if you die you can load the save without having to spend $5,000 on medical expenses and have to drive all the way back to Trevor's safe-house for the Canis and then back to Zancudo.
Assuming you've taken my advice and come in from the east, you're gonna want to gun the Canis down the bridge and build up enough speed so that one, the barrier doesn't send the Canis into a slight wobble (hard to believe, but it's happened) and two, so you've built up enough speed to get a couple of seconds' grace on the alarm. Follow the road around to the right and you'll see a long stretch in front of you, with another barrier and security hut about fifty yards further on and to the left. This is where you want to go, and this is where it gets tricky.
There's a tank that circles around the perimeter of the base. Depending on your luck, it may be heading straight for you from that aforementioned long stretch beyond the barrier and the security hut. The turn in is a BITCH. If you mistime it, you'll hit the security hut and (if the tank is making its rounds, which is sometimes isn't) get blown apart before you can recover. But, if you make it in, gun the Canis between the two massive hangars, which will shield you from the tank and it can't follow you, and you'll come out on the eastern side of the airfield, about a hundred yards from the first of the chocked Titan planes.
There seems to be a collection of soldiers nearby, so you're going to take a few rounds from them. This is why you bought the super-heavy armour. Just keep the throttle full and don't do any hand-braking or any sharp turns that will slow you down. There'll be one of those large flatbed military trucks either coming for you or not too far away. Keep going to the left and try to avoid getting hit by it, but even if you do, the Canis is designed to be robust, and with full armour and a souped-up engine, you'll be able to get the jump on most of the pursuing vehicles.
The beauty about this approach is that there are no more tanks between you and the jet. It's just soldiers and a dozen humvees/military trucks. Keep to the left and gun it onto the open stetch of runway. You'll see a row of three jets one after another, out in the open. Manoeuvre the Canis into position beside one of them and smack L2 and X. It is EXTREMELY IMPORTANT that you do not press triangle before the Canis comes to a complete stop, or Trevor will dive out the door and land in a heap, taking up to ten seconds to fully recover despite the Canis only doing about two miles an hour. You need all the time you can get, because once you climb atop the Lazer, it will take five seconds to get the cockpit open. Hopefully you'll still have at least a quarter of your body armour left, but at this stage it really doesn't matter.
Once you're in the jet, you'll see a hangar straight in front of you and a stretch of road to the right. Don't go right. It just takes away from time that you don't have. Instead, hit the throttle and within a couple of seconds, pull back on the analogue stick and the jet will lift into the air no problem. You'll hear an immediate message warning you that Zancudo is a no-fly zone. Ahead, you'll see a mountain (I believe it's Josiah). Head for it, but veer hard to the right once you've passed the outline of Zancudo. At this stage, the first air-to-air missile will have been fired at you. What you want to do is evasive manoeuvres. You can take one shot and still land the plane on Trevor's runway, but on my third go I never got hit at all because I did barrel rolls and dives to avoid the missiles, and once I was away from the danger zone, I soared into the sky and avoided the helicopters for sixty seconds until my wanted level vanished.
You'd think that was the hard part over, but it isn't. Lining up the Lazer for a landing is a siege. The first time, I lowered my landing gear and started decelerating around the same time I would in the Cuban plane. I missed the runway by a 100 yards and smashed into the ground. WASTED. The second time, the engine failed because it had taken a rocket and the jet nose-dived into the runway. WASTED. The third time, I took no damage from the base and landed on the runway in one piece, only for the landing gear to collapse under me and the plane to stall on the runway. In one piece. That was the lifesaver. Switch to Franklin and quickly back to Trevor, and you have a fully repaired Lazer jet ready to taxi into your hangar (another reason why it's a good idea to have all three opened up and switchable).
It's tough, but doable.