To unlock Eternal Life you need to get 7 extra hearts from using Eternal Heart drops (the white half hearts). There are two different strategies to help make this happen.
Method 1 - Work towards getting an Angel Room after defeating a boss and try to find the Prayer Card active item, this gives you half an Eternal Heart every six rooms when used. To make the Prayer Card more productive try to gain access to shops where possible and buy the 9 Volt and The Battery to get charges much faster, The Habit can also help but as Eternal Hearts are the first to go if you get hit its not as important. Another item to help this method even further is Maggy's Faith which is unlocked by beating The Chest with Maggy, this gives you half an Eternal Heart every time you enter a new floor. The only problem with this method is that finding Angel Rooms let alone one with the Prayer Card can prove time consuming.
Method 2 - Play on Hard Mode, Eternal bosses always (as far as I know) drop half an Eternal Heart after being defeated, you also have access to Eternal Mob Trap Rooms which can provide more Eternal Hearts. While this gives you a great deal more drops it can also be a bit more difficult due to the enemies being much tougher, if you get any Eternal Hearts be sure to complete the room/floor BEFORE picking them up to lessen the chance of you getting hit and the Eternal Heart being lost. Again Maggy's Faith and the Prayer Card if lucky enough to be found in Hard Mode make this go that much quicker.
You do not have to keep the heart containers given by Eternal Hearts so feel free to use Guppy's Paw or Deal with the Devil Rooms at your leisure.