I'm interested especially in the back lever. Some sources that I found suggest it puts strain on your chest. I am not able to perform it, but I imagined that it would be more strain in the back. At least, that would be the case if you wanted to keep the legs straight.
Both front and back levers require a very strong core. By core I mean your entire midsection, not just abs. Your lower back especially needs to be very strong for both these exercises.
In addition to this you'll need very strong lats, shoulders, chest and grip strength for both. A front lever will require a lot more power in your lats than a back lever because you have to keep your entire body up with mostly your lats.
In a back lever your arms are somewhat locked into position because of the way you shoulders work, because of this the back lever is the easier of the two for most people. During a back lever you mainly use your chest and shoulders to keep your upper body in position while using your core to keep your body straight.
If you want to do either of these two exercises you're going to need to do exercises to train different parts of the movement and try to combine them once you've mastered them. For instance, if you can't hold an L-sit for atleast 10 seconds, there is no point in trying a front lever, so train L-sits first. You'll also need to train your upper body (lats/shoulders/chest) to get the upper part of the movement right.
If you have good basic strength from push-ups and pull-ups, you might already be able to to tuck versions of the front and back lever. From there you can transition into advanced tuck, 1-leg-front/back levers, stradle front/back levers. The most important part of this is to keep your upper body straight while holding it. You'll need to record yourself while doing this and look at it afterwards unless you have someone with you who can already do a front/back lever, because it might feel like you're doing it correctly, or it might look cool to by standers who have no idea, but you might be doing it incorrectly without knowing.
Another small tip I can give you, which seems stupid but can really help at the start, is doing training for them without shoes on. The reason for this is because shoes add weight very far from the center point, which makes it harder to keep your body straight. Learning handstands is also easier without shoes because of this reason.
Sorry this kind of turned into a mini tutorial. I hope this answer helps, if you need more information feel free to ask, I've mastered both movements over the years and have also taught many people how to do them.
The back lever is such a good and difficult exercise to perform because it requires strength and balance combined. It optimises a lot more muscles than you may imagine.
Chest, Back, Shoulders, Arms and of course abs. It puts the majority of your upper body to work hence why a lot of people who can perform it have put in the training to be able to perform it perfectly which includes a lot of callisthenics hence they all have lean strong upper bodies