Just Cause

To get the Most Wanted achievements (for 1, 2, and 5 minutes surviving at a Level 5 Wanted Level), you have to remain hidden, but retain aggro so that the wanted level doesn't dimish over the course of 5 minutes.

A good place to do this is here:



Note where the Black Arrow is, which represents the player. It's about halfway south on the map, and all the way east. There's a set of small islands with a bridge there and an unmarked military base. As you approach it, you'll see 3 Red Squares with black filling, which you'll learn are Surface-to-Air (SAM) Missile turrets, so you don't want to go there with a helicopter. Either take a boat or a car to that location.

Once you get to the bridge as you approach that set of islands, your wanted level will jump to 5 and you will quickly get persued. What you want to then do is get down to the beach, which always reduces the amount of enemies that chase you, and press yourself against the coastal wall, to block out the gunfire from the helicopters. You'll continue to aggro the choppers, maintaining your wanted level, but you won't be taking damage from them, allowing you to endure the time.

You may get a soldier or two onto the sand, and if you like, you can cautiously peek out and kill them if they're a nuisance.

I've created a video showing the process.

Check your PDA for a small island about halfway down the map which is connected to the main island by a bridge. There is an army base (signified by three red and black squares on the map) here; drive up to the base and your wanted level will go up right away. Just run or drive around in this area for a bit, and when the police start showing up in large armored cars, hijack one; these things are practically indestructible. Then just drive up and down the road between the army base roadblock and the main highway until the achievement unlocks.

If you're doing the "White Beaches" collection mission one of the packages is right near the army base so it's hard to miss.

I got this by shooting at police after doing a few missions at the beginning of the game. Once I got my wanted level up to level 5, I headed over to a beach away from roads but where I could get back to a road easily if I wanted to. I kept shooting at the helicopters that were chasing me and occasionally went back to the road to shoot some more cops to keep my wanted level up. The helicopter isn't a very good shot so wasn't a big danger and the cops don't come very frequently if you are far from a road. I was on the beach near the bridge to the prison where you break the rebel leader out of jail and I kept near the boundary between the sand and the grass.

Easy.

The second island has a town with a 5 item fetch quest (some drug) in the upper left corner. Just aggro some cops, build up to wanted-level 5 and then run around between the houses. i found dual wielding the infinite-ammo handguns and then quickly taking down any cops while running between the houses makes the chopers hardly hit you.
I actually were able to do another fetch quest in another area with the level still on 5 and then eventually had to start a side-mission to get rid of them, as I would neither die nor get the chopper off myself.

Just got this on accident. I got a level 5 wanted, then grabbed a helicopter. I hung off the back of the helicopter, which began to climb and spin slowly in circles (trying to point the guns at me, I guess). Once the one minute achievement popped, I put the control down and watched.

By the time the 5 minute achievement popped and I jumped off the helicopter, I could see the entire play area. It took forever to fall back to the ground.

Request extraction to Camp Grouper - Agency 02. Head to the dock and use the highspeed boat "Bald Eagle Persuader" head to the right and stay along the coast. You'll pass a town on your right, and then come to a bridge. On the right hand side of the bridge is one of the coca boxes for the collection mission. On the left hand side is an army base, and you will suddenly find yourself on the highest Most Wanted level. A helicopter or two and a police boat will begin firing at you. This next part is critical ... drive in a circle for 5 minutes. Um, that's it. The only complication is that sometimes you might ram the police boat. I couldn't see that this was causing any damage.
Additional notes:
This overlaps with PickAxPete's solution to some extent. I'm sure we are talking about the same island.
After 5 minutes, I decided to hightail it back to Camp Grouper and I found a number of helicopters tailed me. When I got back to the camp, I got out and hijacked one, and then pretty quickly got the 20 Air Vehicle Stunts achievement because the helicopters were so dense that whenever I jumped out of one, I jumped to another.

If you go to the same Island from PickAxPete's solution. You can land on the roof of the army base and just stand there. The three missile launchers will take out most enemy helicopters 90% of the time. While no ground enemies can reach you on top of the roof. If a helicopter is not shot down, you can just stand behind one of the walls on the top, and it won't follow.