Perhaps the easiest way to get this so far is on Avengers international version with easy difficulty settings.
The only move I used during the entire game was the kick (A button by default) you can move around while constantly kicking and kill the enemies by kicking towards them, also if you get grabbed or there are a lot of enemies around then you should use the spin kick move (rotate the left stick frantically while mashing A button).
Its important to save as many lives as possible since you only get 6 and there are 6 short levels. You can earn 2 lives back through scoring points (at 20k and 80k). It helps a lot to destroy the power up objects that look different on each level, these can give you a speed boost, extra points, increased power and most importantly health!
Some of the enemies are a pain and it will help to memorise where certain enemies spawn to help kill them instantly or avoid them totally and therefore avoid unnecessary damage!
Below are the strategies I used during each level and the boss fights.
"South paradise city"
Work your way up the level killing the red enemies and destroying/collecting the power up objects, be aware of the bomber guys, be sure to avoid there 3 bombs before they run off. Try to take out the boomerang enemies instantly which spawn out of gaps on the right. There will also be quick ninja like enemies which will need your attention, either take them out mashing A button while pointing directly at them or avoid there downward path completely.
When you get to the top of the first section move forward carefully and stay on the right side until the bomber guys back is against the wall, take him out approaching him from the right hand side. Next time between the gunner in the windows bullets and take him out with some diagonal kicks and progress to the end of the level by killing all the red enemies at the top and dealing with the individual enemies accordingly on the way.
*Ball and chain boss*
If you have full health its a breeze, just time between his ball and chain spin move and then unload about 3-5 kicks into him each time, back off until his spin move stops then unload again, rinse and repeat! You should do a lot more damage than he does, especially if your careful to avoid his attacks.
Instantly head right and quickly move forward on the extreme right hand side, this will help you avoid the assault from the sai throwing ninja's, if you get grabbed just do some spin kicks, you may take a couple of hits but don't worry as there is some health on the left right after the first sai throwing ninja section.
Deal with the quick downward enemy near the top then head right around the corner, there will be more health here. Next move forward quickly staying on the extreme right again all the way to the top. Be careful of the guy in the water throwing projectiles and dispatch all the enemies to progress to the boss.
*3 red sword guys*
Instantly head up to the top left corner, squeezing imbetween the red sword guys on the left and middle while they are flashing. Once in the extreme top left corner constantly tap X and A simultaneously to activate some spin kicks. Be patient and eventually you will take out the true boss without taking any damage. Credit goes to wariomona for bringing this to my attention.
"North paradise city"
Progress forward in the usual way, kicking out at all the enemies, there is a new enemy which is in the ground but he is easily avoidable, just walk round him. When your approaching the middle area there is another guy in the water throwing projectiles, just time imbetween the projectiles and move up safely NOTE in the next area is another guy throwing projectiles but he can change the direction he fires, take caution and avoid his firing lines at all costs while moving over to the left.
Keep progressing upwards dealing with the individual enemies correctly until the next boss fight
*Big blue projectile boss*
Instantly break open the gate at the bottom and avoiding the blue projectiles head to the top, you want to be right next to the boss on either side, once your alongside him you can just literally kick him from the side until he dies.
Move on up staying away from the cliff edge on the left (dropping off is instant death). Kill the easy enemies on the way, at the first corner near the cliff edge is a strength power up. Turn the corner and there will be a boomerang guy on the ledge, move up past him quickly and carefully being sure to kick any boomerangs coming your way.
In the next section there will be rocks falling from the right, just take out anything that's getting in your way ASAP I.E the rocks, enemies and boomerang guys, move forwards and take out the bomber guy when his back is against the wall approaching and attacking him from the right like before.
When you head all the way left there is some health, now kill all the easy enemies and your at the next boss.
*Bridge kicker boss accompanied by bomber*
Try to stay near the bottom of the bridge and lure the kicker guy down low to keep away from the bombs. Try to time it so your moving up towards him a little to lure his kick move, then back off until the second he stops the kick and retreats, the second he begins retreating then instantly follow him up and try to land a few upward kicks on him, NOTE it will go orange if you've landed a successfully hit and sometimes if you hit him with more than one kick he will likely counter so be careful not to overdo it, however you'll want to make sure the timer doesn't run out either as you will lose a life.
"The forest of treachery"
As the level starts head right then up and take out the sai throwing ninja and get the speed boost power up from the skull behind him. Another sai ninja may follow so be cautious and kill or avoid him. You'll want to keep and eye on all the green snake things which fall down during the level, some accurate well timed kicking will do the trick against them. Another stationary sai ninja is nearby, just avoid the path of his fire and take him out.
Around the corner the next area is quite tricky, when you start moving up there will be lots of sai throwing ninja's which jump across the screen throwing a few sai's! I just progressed forwards asap killing anything that got in my way, its better to just run through this section and maybe lose a life as taking it slow won't work in your favour healthwise.
2 stationary sai ninja follow's, same goes for them as the others, avoid there path of fire and squeeze past the first one around a corner, then head right then up and take out the second one attacking from underneath him. Next head right and edge out (being careful of the green snakes) far enough so that you've triggered the last enemy sequence then head back under the rock on the left where your safe from the snakes. Take out all the enemies in this small area using spin kick if needed and pick off the remaining enemies one by one that are among the snakes, returning to safety under the left rock.
*Jumping water boss*
Head up near the waters edge, be sure to try to take out as many green snakes as possible. When he jumps out let him land and get fully on the ground before you start kicking at him from the side and kick diagonally as he retreats back into the water. It can sometime be tricky to hit him but occasionally he will fall down and turn invisible after recieving enough kicks, this is when you will want to unload as many kicks as possible, rinse and repeat.
"Palace of the gashita"
Move steadily upwards taking out all the blue enemies asap attacking them from below. When you reach the doors kick the lock to open one of them and progress up making the blue boomerang guys your priority, and kill them quickly to avoid unnecessary damage. The last flaming skull on the right hand side is health so grab it if needed then take out the standard enemies before the last boss.
I was lucky and made it here with 3 or 4 lives left and just unleashed as many kicks as possible literally toe to toe! If your low on lives and health another tactic is to stay out of his kicking reach and when he keeps moving left and right, time it so you make to the top middle bit next to the throne/chair. He will now be trapped in the corner and if you slowly edge down and around the corner (being careful not to go too far downward or he will go for you) you can land 2 hits on him before he counters (you will need to move out of the way asap after the second hit connects) some of the time he will block your kicks too so be careful to only land 2 kicks then move away. If anything goes wrong back off down the bottom and wait for an opening to reach to top section again, rinsing and repeating the same strategy.
I got really lucky when I popped the achievement, just totally went on the rampage killing and collecting everything! Good luck to anyone going for this, if anyone needs more help feel free to comment or message me. Hope this helps out.