The ultra rarity of the trophy might make this one seem intimidating at first, but in fact it is very simple to achieve on easy mode ("I'm Too Young To Die.") I strongly recommend completing the campaign on normal ("Hurt Me Plenty") difficulty first. Doing this will familiarize you with combat to the extent that maintaining a healthy stock of extra lives on easy mode should be no problem.
You're obviously going to want to collect as many extra lives as possible during the playthrough. Some upgrades won't hurt but maxing everything out is by no means a must. I advise against attempting all secret encounters and slayer gates - the reward isn't worth the risk. All collectibles should be obtained in your first playthrough if you're going for the platinum. I ended the campaign with almost 50 Extra Lives even with making mistakes here and there.
Trophy pops after you land the killing blow on the final boss. Happy hunting!
P.S. - Same method works well for both DLC trophies requiring you to finish the chapters with 5 or more extra lives.