I had a copy pasta made for this, but I guess I can't find it. So here it goes again.
There's a Civ5 mod called Acken's Minimalistic Balance which is aimed at (mostly) making AI changes for single player. The mod does the following major changes:
>makes the AI much smarter
The AI now plays to win, while still adhering to the leaders' different personalities. I've had a game where I went for a Diplomatic victory and the Swedish AI just started taking over every city state it could in order to stall me. I've had ultra unfriendly leaders go from War > Peace > Friendship > Joint War Declaration in 2 turns because another AI won a war and gained multiple cities as part of its peace treaty. At the same time, I've had friendly leaders be bros for the entire game as long as I didn't piss them off. It's a big change from the base game where the AI is generally just sitting there and waiting for you to take action.
>melee unit and city combat strength rebalance
Melee units are extremely deadly. An AI doom carpet will take an unprotected city in 2-3 turns with no casualties. You can no longer play tower defense and watch the AI suicide melee units into your archer/city range. The defensive strength of ranged units has been massively reduced (7>4, 18>11, etc), the melee combat strength of melee units and cavalry has seen some boosts, and cavalry get a lesser penalty for attacking cities. Early/mid game ranged units just get 1 shot by equivalent tier cavalry and iron melee units. Even barbarian horsemen will 1 shot your archers. It completely changes warfare, which in turns completely changes build order, unit production, maintenance, AI willingness to declare war, etc, etc. Melee ships are also changed: they can now attack and move by default. It does make ships a bit too strong though, so I'd recommend digging through the files and manually nerfing the melee ships by like 10% or so.
>religion and ideologies rebalanced + tall nerf/wide buff
Tradition is no longer the absolute best start in 95% of cases. Piety, Honour, and Liberty are now extremely viable; and as a bonus the AI no longer gets hobbled by picking those. Many tenets are buffed or changed, smoothing out the usefulness of the less desirable beliefs. On top of that, the tech and culture penalties for going wide are significantly lessened, making Liberty/Honour/Piety that much better.
Overall, the game DOES get significantly more difficult, but it makes the Civ5 experience way better. I had to go from stomping the vanilla AI on immortal to struggling with the modded AI on King. I also had about 200 hours into the base game and was starting to get bored, but with this mod I'm now at 2600 hours. It basically turned Civ5 into a "forever game" for me, where I can always go back every few months or years and have a great time playing a few games. Kinda like HoMM3.
There is one change the mod does that I modded out though: the mod changes the damage taken value by embarked units from 200% to 50%. It effectively makes embarked units able to beat melee ships 1v1. I changed that back to it's vanilla 50% value. The AI will still gladly suicide its units by embarking them near your cities, but that beats having 2 triremes struggling to destroy a horseman on a raft.