This solution is going be very long winded but, it'll have a lot of useful hints and tips to make the achievement that much easier to get. Please read everything and leave a reason in the comments as to why you have down voted the solution, if you have.
This is the 2nd to last licence you're going to earn in the game. For this, you'll need to complete every single event on the main island of the game. There are five types of events:
1. Races -
402. Stunt Runs -
143. Marked Man -
154. Road Rage -
165. Burning Route -
35 -
IMPORTANT NOTE: DO THESE EVENTS AS YOU UNLOCK THE CARS. YOU'LL ONLY NEED TO DO 95 EVENTS FOR YOUR ELITE LICENCE IF YOU DO!!!!The numbers in bold represent the number of events per type of event. Each one
MUST be won in order to get a mark on your overall licence. Each licence upgrade will also reset all events on the map with the
exception of Burning Route events. Once these have been completed, you'll unlock an upgraded variation of the car you used in the Burning Route event and keep that car.
If, at any point, you fail an event and want to replay the same event, simply press

on the D-pad and scroll down to
Restart Last Event and press

again to restart it. This will save you a lot of time driving back and forth. You can also use this as a fast travel by restarting the previous event and stopping your car. This will cancel the event but, leave you in the same spot and allow you go to near by events you haven't completed yet.
In order to get to the stage of acquiring your Elite Licence, you'll need to earn each of the other licences. Starting from your learners permit, you'll need to
win 2 events to get your
Class D licence:
Once you have this licence, your events will reset and you'll need to
win 7 events to get your
Class C licence:
Again, events will reset and you'll need to
win 15 events to get your
Class B licence:
When the events reset again, you'll need to
win 26 events to earn your
Class A licence:
Second to last time the events will reset and you'll need to
win 40 events to unlock your
Burnout Licence:
Once the events reset for the final time, you'll need to
win all 120 events to unlock your
Burnout Elite Licence so, lets get on to some tips.
Races - 40 To Complete:: These are straight forward. Drive from point A to point B and be the first one to cross the finish line. I would recommend
Stunt boost types cars for these events as you'll be able to use a lot of the environment to build up your boost. Getting takedowns on opponents early on in the race will also help.
Stunt Runs - 14 To Complete: Simple objective for this one; combine various stunts and keep your chain going for the duration of the event to hit the target score. You have 6 ways to get increase your multiplier;
Big Air (x1),
Billboards (x1),
Super Jumps (x1),
Flat Spins (x1),
Burnouts (x1) and
Barrel Rolls (x2). If you hit a ramp just right, you car combine a flat spin and barrel roll into one stunt however, levelling out the car when landing one of these is a bit tricky. You'll need to have great knowledge of the map in order link up multipliers. Using your
Boost (

) and
Drift (tap

and turning left or right) will help keep the chain going longer. Additionally, there will be a white circle at the top middle of the screen that will slowly fill in red. When this fills up completely, the combo is over. Even when you run out of time, the combo will keep on going until you lose it by crashing or by not drifting/boosting. The obvious boost type car recommendation for this one would be
Stunt. If you go into
Legendary Cars from the junk yard menu and pick the
Jansen P12 88 Special, you'll have a much easier time in completing these stunt runs. Very fast, easy to turn, great at drifting and can nail double barrel rolls on the right ramps.
Marked Man - 15 To Complete: These are, in my opinion, the toughest events in the game. Your objective for this one is to get form point A to point B, without being taken down 3 times. If you use a strong car, you'll be able to take 4 hits before it's event over. The AI will be using the
Hunter Civilian car. You can either use this car to even the odds or, pick a different aggression car that suits you.
Aggression based boost cars are your best friend for these events as they're the strongest cars in the game. This will make it a lot harder for the AI to take you down. By the time you start going for your elite licence, you will have unlocked enough cars to have a wide selection of cars to pick from. (
Side Note: If you start Big Surf Island DLC and do the burning route for the Dust Storm Buggy, you'll unlock the Dust Storm Buggy Super Turbo. By using this car, you'll be put up against the regular version of the buggy which is also a big advantage.)
Marked Man Update: GrmRpr69 points out in the comments that the Super Buggy tip doesn't work anymore. Seems that it has been patched by the developers. Thanks.
Road Rage - 16 To Complete: Your objective for these events is to take down a set number of AI controlled cars. When you go for these events, you'll be tasked with taking down at least 20 vehicles. These will get very annoying, very quickly as the AI will get stronger and harder to take down.
Aggression based cars are, again, you're best friends for these type of events. You'll gain a significant amount of boost back when you get a take down which makes it a lot easier to catch up to the AI to get more take downs. Do not worry if you take a wrong turn and lose the AI, they will automatically spawn behind or in front of you.
Burning Routes - 35 To Complete: For these, you will unlock cars after a certain number of events have been completed. These cars will then free roam the city until you perform a takedown on them. Once taken down, they will be transported to your junk yard. From here, select the new car that you've unlocked. Once you're driving the car, head to the nearest repair shop (log is a white spanner in a blue shield with a white outline) and get the car fixed. This will automatically identify the location of the Burning Route event for that car on the map in the form of a
white star in a red circle. Unfortunately, you're unable to mark the event on the map and get GPS guidance to the location so, you'll have to make your own way there. If you're lost, simply press the

button to bring up the full map and find out how far you are from the event. The star will still show up n your mini-map but, you'll have to navigate to it by creating your own route in your head. Once there, simple press and hold

and

to start the event. the event is simple enough; get from point A to point B within the time limit. your reward for each of these will be an upgraded version of the current car you're using. Once you complete your final Burning Route event, you'll also earn:
That's everything I can think of to help makes this achievement somewhat easier for you to earn. If you have any questions, inbox me or leave a comment down below. If you leave a negative vote, please state why and what could be done to improve the solution.
Happy hunting and best of luck!