Being a Nightblade vampire, in full medium armor, with rapid maneuver (or speed potions), and maxed force lock chance helps. I usually get them done with quite a bit of time to spare. Basically look for ways to increase your movement speed while in stealth and anything that helps you avoid detection.

Speed pots are not the best option in my view. My win/win combination is wearing a set that removes stealth speed penalty, another set with stam regen, jewelry with potion boost and potion cooldown reduction, and invisibility pots.
With this equipment, everything becomes quite easy. Being a nightblade is another asset, but it works with all classes, and you don't have to be a vampire at all.
You'll still have to learn the mobs' positions and routines, but that comes quick. You should be able then to complete all heists, no matter the combination, with 1 to 4 minutes security margin.

The easiest way, as others have noted, is to be a Nightblade. They get both invisibility and fast sneaking built in the class itself. The next easiest is way is to be a vampire. Combine the two for an overkill.

Now supposing you want to do the achievements on a character that is not a Nightblade, and don't want to go vampire, then there's a bit of prep involved.

I went through Thieves Guild with a magicka Templar who is not a vampire, so while doing my perfect run achievements, I learned a trick or two about how to do the heists.

Get a big stack of potions with speed and invisibility. Also get your alchemy high enough to get the potion duration buff passive. You want to maximize both the time you are invisible as well as the speed buff. Investing in potion jewelry is also an option.

Next get some sneak detection reducing gear. 3 pieces of Night Terror and 5 pieces of Night Mother's Gaze do the trick. (Being a Bosmer works too.) They are medium sets, but it doesn't matter, even as a magicka build, since you will not be doing any fighting during heists. If you do, your chances to get a perfect run are slim.

Another option is to get a set that removes the sneak movement penalty, like Shadow Dancer's raiment or Night's Silence. You can combo them with a sneak detection reduction set too. Note that Shadow Dancer is a light set and Silence is crafted. Ideally you want to get at least 5 pieces of medium armor for your sneak gear.

Level your Medium armor skill line and get Improved Sneak passive. This is the reason you want to wear medium armor. Even if you are otherwise magicka, or usually play as a tank in heavy armor. You will be amazed how blind the NPCs are with all of these sneak detection range reductions combined.

Finally, it's a good idea to get some sneak cost reduction in addition to the medium armor passive. Legerdemain and Shadow CP tree both have it.

With that done you can start to learn the heist maps. It's a good idea to go each map over once like a regular dungeon. That is kill everyone, explore everything, check what paths are accessible, where you can climb and jump to and so on. Keep an eye out for hidden paths and shortcuts. You will get a bad review for such a heist, but the information you gain from doing that is crucial to understanding how each map is laid out.

Tackle each map in the same way each time you get it. Pick a path and get used to it. That way it will be easier to remember where the traps are and what the guards do. It's possible to make tight loops in each map that have many possible target chest spawn points along it.

Learn when to quaff a potion in each map. You need to know how to pace your potions usage. Some guards are easy to avoid by sneaking, with others you need to wait for them to go away, while other spots are too congested for it to be sensible to wait for an opening. That's when you use a potion. Ultimately it is a matter of experience when to drink one.

Note that there is a bit of luck involved in each heist, since the stuff you are gonna steal is randomly distributed in the map. Also some maps are easier than others. Sometimes the targets spawn in far away corners of the map and getting to all of them undetected takes too long.

You can game the system by abandoning the quest before leaving the heist location, logging out and re-logging to reset the timer, and trying your luck again, with hopefully a bit better selection of targets.

Finally... Traps. Do not bother with bear traps. Disarming them pulls you out of stealth and that can be often a bad thing. The rest you can disarm or hop over. Staying in stealth makes it easier to spot traps, so sneak as much as possible, but also keep in mind that many maps have long vacant corridors and it is prudent to sprint while traversing them. Every second saved is precious. Learn the location of traps on your selected route. All it takes for the heist to go awry is to miss one trap. You stumble on a tripwire, a guard notices you and then it snowballs from there.

So doing successful heists is a matter of running a Nightblade, or being properly prepared. In both cases you need to memorize the maps and develop your standard route for each one, and keep doing it until you get it down to science.

Once you are familiar with each map, and have the proper gear and the required skills, then doing heists becomes pretty easy. Just keep in mind to move as fast as possible, and go straight from one objective to the next. Do not dawdle and do not stray to do other things like picking pockets or going through containers.

I play as a Magblade but I have avoided the Thieves' Guild quest line altogether. I watched my BF try the beginning quests and become more and more frustrated. I absolutely hate being timed and when playing games avoid the ones that time you. Even though my character is probably perfectly set up for it, I don't want to bother. I used to play as a stamina character and so my medium armor skill is high. My alchemy skill line is maxed with all the passives purchased. I ditched all the medium armor passives when I switched to a Magicka character but I could get them back. Even trying out the Dark Brotherhood, which at first I enjoyed, led to frustration when I was given a timed quest and told not to let the NPCs in the location see me. So I guess that's out. I admire anyone who has the willpower to tough these out. The timing and stealth requirements are too much of a bother for me. Good luck to anyone who tries though! You're going father than I did!