If you look at guides for character classes in computer role-playing games and MMORPGs, you will often find theorycrafting advice to optimize damage per second or similar parameters. In a tabletop role-playing game you could probably do something similar, but as fights are less frequent, less repetitive, and less predictable, the mathematical theorycrafting doesn't always work so well. And because your character will do other things than fighting, you'll also need to look for other aspects, like whether his class / race is interesting, or what non-combat skills he has.

But if you look mainly at combat, what makes a D&D character fun to play? The answer to that question is usually that a fun character to play has interesting and useful things to do in different situations. While you can't possibly plan for every conceivable situation, what you can do is to have a look at three relative common situations:

1) The all-out, alpha strike attack: This is when you don't care about conserving resources like spell slots, but want to bring down an enemy as quickly as possible. This would for example apply if you happened to come across a boss fight after having rested. So when you look at a character class or specific build, you need to ask yourself what you would do in this situation. Is there something you can do that will really stand out, like a very powerful spell or combination of spells and other tricks? I once encountered a dragon with my level 5 warlock, who had Hex already cast from before a short rest, in which he recovered both of his 2 spell slots. So I used my spell slots to cast Scorching Ray (from the Fiend Expanded Spell List). As I was casting all spells at level 3, I had 4 Scorching Rays instead of 3, and because of Hex they did 3d6 of damage instead of 2d6. So in two rounds I was able to cast spells for 24d6 of damage, which really contributed a lot to us vanquishing that dragon. This sort of gameplay is fun because it is flashy, but you can't do it very often.

2) The other extreme is what you do when you have run out of resources. There are two things to consider here: How likely is it to run out of resources, and what can you still do when it happens? Low level spellcasters are quite likely to run out of spells, and warlocks have that problem even at higher levels. But then a warlock has the best cantrip in the game, and if he still has Hex up, and has the Agonizing Blast invocation (which he usually should), his Eldritch Blast is powerful enough to not feel boring. The most bored I ever got with a character class was with a lore bard, who tended to cast only one powerful, concentration-based spell, and then had to cast Vicious Mockery every turn for 1d4 damage, because his other spells needed concentration as well and would have broken the first spell. Melee combat characters usually are comparatively good in fighting without special resources, and thus can go on and on.

3) The third situation is the regular combat, in between the first two situations. You want to conserve some resources, but you want to do some interesting stuff. A spellcaster might for example use lower level spells. A barbarian might decide that he has enough rages per day to use one for a standard encounter, but won't be using Reckless Attack. For some classes there isn't much room between all-out and no resources left, because they only have so few resources to work with. For other classes, like higher level wizards, there are still quite a lot of different options.

So when I have to choose a character class for a new campaign, and look at some class that I might be interested in, I do try to imagine the above three situations. Do I have actions available in each situation which are at least somewhat interesting and powerful? What constitutes interesting might of course vary from one person to another, which is why I won't make a list of "best character classes". But if you have a plan for heroic fights, very long fights, and standard fights, chances are that you will have fun with that character.