Like what GlassJaw posted, I'd advise heavily considering the power of individual feats and considering their value individually, and not just adding a blanket cost to feats if you're looking to increase the amount people can grab. I see you plan to "split the feats in two", but I'm not sure what you mean by that, so I'll just toss a random point to consider on that note.
I haven't personally thought about this a lot, but what immediately comes to mind is I'd split the feats into 3 categories: those that affect damage output directly, those that grant an ability score increase, and then everything else. You can grant an ASI at every 4 character levels as opposed to class levels; however, if you gain an ASI from an actual class, you swap that entirely for any feat that has a single ASI built in. This way classes that get ASIs, and people that opt to stick to one class anyway, still gain something at their ASI levels, but it's not an entirely free selection to stack on top of other free selections. The feats that give an ASI tend to be things like gaining a proficiency in a save (Resilience), some form of armor proficiency (Lightly Armored), a significant boost to a skill (Observant), etc. They don't tend to be things that can give a significant combat damage boost, like Crossbow Expert, Sharpshooter, Polearm Master, Great Weapon Fighter, etc.
I find that the extra attack and damage increase feats tend to be the most powerful when players are able to stack them, so I'd be less reserved in handing those out. If the feat somehow gives you extra attacks, or increases your damage directly, it falls into this category. You don't want everyone to be the same because they can all start out with Crossbow Expert and Sharpshooter right out the gate, or all start with Polearm Master and either Great Weapon Fighter or Sentinel. Everyone will likely want to start with those, and the people that don't are going to very rapidly fall behind in damage output, making balancing encounters quite difficult with such a heavy disparity. For this reason, those feats perhaps should require the player to spend their actual ASIs at every 4 levels to obtain, as opposed to being able to grab for free in some way.
The "other" category of feats, AKA the ones that give neither an ASI or direct attack increase, can probably be given out a bit more openly, something like every 5 character levels, or whatever you want to come up with. These tend to be feats like Dungeon Delver, Grappler, Healer, Inspiring Leader, etc. I feel that these ones are the ones that add the most extra flavor to the individual characters, and getting one free every 5 levels or so should not be overwhelming. If you want you could maybe give one of these out at level 1 as well, just so they can start with that little bit of extra personalization, but I'd weight whether or not you feel you're giving people too much first (maybe make some test builds and see what you can do).
With all of this, ignoring a free level 1 feat, a character will be the same as normal until level 4. At level 4, they get the normal ASI, even if multiclassing. If they didn't multiclass, on top of the normal ASI they get a free feat that offers an additional ASI, such as Observant or Resilience. At level 5, everyone gets a free feat that does not give an ASI nor give extra attacks or damage output, such as Healer or Tough. If they're multiclassing all willy-nilly, they may not see any extra benefit until level 10 after this, but they'll still get their full ASI at 8 regardless, and another free "other" feat at level 10. People that stick to a single class that gives extra ASIs like Fighter or Rogue, should maybe still be allowed to spend the extra ASI levels (meaning not 4, 8, 12, etc), and grab any feat they want, since it is a class feature that would normally not be limited; however, if you do feel it OP to do so on top of all of the free feats they could get, and the free ASI feat at level 4, maybe limit these to one or two of the feat categories as you see fit.
I just said all of this as it came to me, so maybe it's full of holes, but it's hopefully at least food for thought.
