Try uploading a replay so we can see what you are doing wrong. Normally spamming light frigates is enough to get you through hard AI.

I hope this isn't a case of forgetting to upgrade planets and having negative income the whole game....

Basinator, if you have autorecord enabled, they will save into C:\Users\[YOURUSERNAME]\Documents\My Games\Ironclad Games\Sins of a Solar Empire Rebellion\AutoRecord-SinglePlayer

For manual recordings, it's a different folder at the end: Record-SinglePlayer

WOEaintME's recommendations should get you up through hard AI, so long as you are keeping up with your economy--make sure you don't have planets with no infrastructure,* always build your extractors, and don't research the next fleet upkeep increase until you are running out of fleet supply.

There is also an in-game tutorial that goes through some of the basic mechanics, in case you are missing something easy but not obvious to a new player.

*Civilian infrastructure - select the planet/asteroid, click the green planet upgrades button in the planet's control panel, then click the button for civilian infrastructure (it will prominently feature a person's silhouette). Make sure that you upgrade until the credit income is not negative (you can see a planet's tax income by hovering the mouse over it; you can also see how much income it will have by hovering your mouse of the civ infrastructure button). For dwarf planets and asteroids, this almost always requires only one upgrade. For larger planets, this almost always requires two upgrades.  Note that additional upgrades will allow more population, eventually allowing you to gain more money from the planet.

Basinator, if you have autorecord enabled, they will save into C:\Users\[YOURUSERNAME]\Documents\My Games\Ironclad Games\Sins of a Solar Empire Rebellion\AutoRecord-SinglePlayer

I don't have any Ironclad folder there. I am using the main game, the very first Sins only.

Basinator, I've loaded up your replay. Some of my instincts are going to be distorted because this is original sins and because it's on a slower speed than I'm used to, but I have a bunch of feedback, in the order that I'm noticing things:

1. Even in original sins, the Kol is not a great starting cap. Use the Sova or the Akkan.
2. You can right-click the move button (a green arrow) to set your ships to "All together"
1. This is helpful to avoid sending in a lone light frigate, or whatever, first/
3. Good micro on your colonizing fleet.
4. Consider getting the terran population upgrade earlier, to increase money income
5. Upgrade your civilian infrastructure on new planets--usually, before you buy anything else. The exceptions should be emergency situations, e.g. you have to immediately defend the new planets or spam fleet elsewhere. Your first desert was negative for ~5 minutes before you started upgrading, cutting your money income by almost 50%.
6. Don't research volcanic occupation when there are no volcanic planets on the map
7. Push towards a choke point first, rather than taking worlds that don't deny your enemy access to anywhere else. E.g., you took an ice world for your second planet that was 4 jumps from an enemy at a time when a desert or a closer ice world would have been better. This can be hard to figure out when you're surrounded, but just bear it in mind as you gain experience.
8. Related to 7, prioritize planets that don't require your fleet to backtrack. That ice world took your fleet 3 jumps, when you could have gone one jump to a desert choke.
9. Be ready with either fleet or defenses earlier. I am currently watching at the 18 minute mark, and you only have a Kol and 6 frigates (while the AI have more than 15 disciples). Either fleet up and get ~20 LF (preferably LRM), or research repair bays and put them down on your choke points with gauss.
10. In early battles, you usually want to focus fire the enemy frigates--unless you see an opportunity to actually kill a cap ship. (Also, a repair bay or phase jump inhibitor will really give you a defensive advantage.)
11. When you're going to lose a fleet battle, run away--preferably before all of your frigates are dead.
12. At the 40 minute mark, you are trying to recover. Build an extra frigate factory or two to spam ships better--you're sitting on a lot of resources that could be ships defending your planet, but instead your fleet is tiny and a planet is getting bombed out.
13. At 47 minutes - Frigates are far more effective dps per fleet and per resources than capital ships. Since you are fighting two players, you have to be efficient.
14. In terms of cap ships, you usually want diversity--a second Kol is not going to bring any new, helpful abilities to the table.
15. At 50 minutes, you went off to defend that far ice world instead of taking back your close desert--this is why your hw was vulnerable.
16. You almost always want ships of a type to focus fire. At 56 minutes, your cobalts are spread out and hitting different enemies, so you end up killing fewer ships than you would have.
17. Watch your cap ships; you jumped a high level Kol with a third of its health into a battle against overwhelming numbers near 1 hour.
18. In an ongoing fleet battle, building repair bays should be a priority over gauss.
19. At 1:14, you're again sitting on resources when you should be spamming frigates.
20. Generally, your opponents were running almost exclusively disciple vessels. These are the Advent light frigate and so they are extremely weak to Javelis, the TEC long-ranged frigate.
21. Generally, if you're playing defensively, you should turtle in a close cluster of planets to avoid a lot of jumping your fleet around. On this map, you probably could have held up on the close ice world and the first desert to outlast your opponents.

I hope this was helpful. You clearly have most of the fundamentals down, so I'm not surprised you won the next match.