This is no easy feat at all and is probably the hardest achievement to get. Hardcore mode only grants you 1 life at the beginning and your best bet is to play through in Single Player and collect everything whilst beating everyone up. There are at least 5 extra lives hidden around the game and they can be found at:
Level 1 - as soon as you walk onto the screen go to the bottom left corner and attack the wall.
Level 2 (the Bridge) - After you exit the fight in the truck, go back behind the truck cab and it can be found by attacking the top left of the screen.
Level 3 (Pirate Ship) - there are 2 crates in this mini section; 1 has an apple inside, the other has the 1up.
Level 3 (Ghost Train) - When you approach the fight with the monster that comes out of the wall but before you trigger him, there is a 1up under his head at the very bottom and in the middle of the screen.
Level 7 (Factory) - There is a 1up in a Barrel about half way into this level. Just destroy everyone you find until it appears.
Also, you gain an extra life at 20,000, 50,000 and every multiple of 100,000 from then on.
Plus look out for some different named characters like V-3 etc these guys give you extra points for killing them.
Tips for killing the bosses:
Level 1 (Barbon) - If playing as Axel, this guy goes down quite easily to Grand Uppers, just dispatch his cronies before triggering him. Otherwise, he's much more aggressive than you'll be used to, but his throws should be no threat with your recovery ability, and you're still more skilled in a ground fight. Watch for blocks and throw for serious damage when you get a chance.
Level 2 (Jet) - Keep the heck away from Jet. His grab attack is nothing to scoff at, and you won't have time to react to his charges if you're close to him. Preferably, stay at the opposite side of the screen and jump kick him whenever he makes a move on you.
Level 3 (Zamza) - A really cool character and one I always wished I could play as. Again, if using Axel he goes down easily to Grand Uppers and grab combo's. If not using Axel, your biggest problem is that he is a very mobile target and likes to lunge across the screen quite a lot. Grabs are mostly useless, as he'll follow up with a suplex of his own right off the bat. Jump kick on any attack except his upper lunge and Special finish him if you've got health to spare.
Level 4 (Abadede) - You can get owned or lose a lot of lives if you try and get too close to this guy as combo's and grab combo's will only get you hurt. With Axel your best bet is to do quick, high damage moves like Grand Uppers or Tiger Punch. Using anyone else, I found jump kicks especially useful since it's tough to go toe-to-toe with this guy. Always dodge the lunge; he's just too high up to hit with your jump kick. Finally, as with any attempt on this guy, do NOT attempt to grab him unless you are absolutely certain it will work AND that you will throw him immediately afterward.
Level 5 (R.Bear) - A serious headache. Make sure as many of his minions are dead before you actually set him off to keep the battlefield clean. Watch out for the lunges and the vicious headbutt grab. Do not hesitate to use specials if he's got you clean and abuse any weapon you have if you happen to have kept it thus far.
Level 6 (Souther & Stealth) - Ok this is a very tricky combo to face. Try and focus your attention on Souther as killing him will drop the Jet clone as well. Where as taking out Stealth will still leave you with Souther. Use the same techniques as killing Zamza.
Level 7 (Particle & Molecule) - This fight can either be a bit of a break or somewhat annoying. Neither robot's attacks do any deadly amount of damage by themselves, but they have a rather large range on their swing, are always jumping around the screen and are generally trying to distance themselves from you. Make your grabs fast or they'll give you a shock. Also, don't let them hover around low health for too long or they'll blow themselves up near you for a cheap kill.
Level 8 (Boss Rush) - The lift is small, so there's not a whole lot of room in your fights. The boss order is the Stage 1 boss, then Stage 3 boss, then Stage 5 boss (Different names, of course). You'll get a solid health boost every section of the elevator, so try not to lose any lives near the end of a fight. Refer to the above strategies, although your biggest problem will likely be the Stage 3 boss combined with an Electra.
Level 8 (Shiva) - One of the most unfair fights to be sure; he counters your combo after the first hit lands, blocks whenever he feels like it, and has some seriously damaging options at his discretion. Don't bother with combos, as he'll flame you in the face after the first hit connects. Mind the time limit, don't hover for too long, and take any open opportunity. Specials are the absolute best strategy if you have it to spare.
Level 8 (Mr. X) - The final fight could be worse. His machine gun spray is a wide 90 degree angle, but it doesn't do too much damage if you happen to get caught. Your biggest concern is his rifle butt, which does a harsh amount of damage and puts you away from him. Time your grabs, never get surrounded by his minions and avoid the back end of that rifle at all costs -- you're going to be hurting for lives at this point and you cannot continue, no matter what.
The general strategy is to smash everything you can, kill everything you see and save often to avoid a huge headache from failure. The 800,000 comes at the very end of the game, so you may expect yours thereabouts unless you clock faster times. Good luck.
Note: If you just can't complete it on Hardest even with save abuse, there's a good strategy mentioned in the Roadmap. To recap:
Start a game on Hard difficulty with 8 lives. Play with the "kill everything, collect everything" mentality until you're about to hit 800,000 (just short about 5,000 will do). Save your game and exit to the menu. Start a new game on Hardcore, and play for around 30 seconds. Exit, and immediately load your Hard save. Get the rest of the points you need, and it should click. Nice!