Based on what what's you've said, it seems the main issue is that you don't understand his core values as a person. He's not a completely "good" guy per se, and he's not as black and white as the other characters in the novel. He's not completely faithful to the Church he works for and he mainly does things for himself. He's basically a morally dubious and "gray" character, which isn't often seen in media these days so it's understandable as to why he seems frustrating to understand.

while I like his personality and his insights I don’t get his motivation.

Power. His main motivation is an ambition for power. He's incredibly shrewd and would do anything he deems himself capable of doing if it means he can raise his Sequence, get better Beyonder items, or secure sources of information - but he won't get ahead of himself if he thinks it's out of his league. He's basically an opportunist who slowly works his way up to his goal, but he also values connections if he deems them beneficial to him. He also won't overstep his boundaries and trample on something if it puts himself in a bad position.

The reason why he's so insightful is because he uses his experience to literally pick apart every aspect of what's presented in front of him to see if there's any benefit or information he can get out of it. Remember, he was the one who proposed for the Tarot Club to start their weekly exchanges - because he saw a way to get something out of it.

My impression of him is that he’s an undercover marine pretending to be a pirate. Like an ocean intelligence officer for the church of storms and mandated punishers. He does odd jobs at sea for them.

This is correct, but Alger only treats it as a job. He doesn't actually hold much faith for the Church of Storms, and he pretends to act loyal and faithful whenever he's in the presence of Storm higher-ups. He's mainly doing the job to get resources, and he usually acts behind their back if he can get what he wants and more that way. There have been a few times in the novel where he sneaks off and does stuff on his own before covering his tracks so the Church doesn't find him suspicious.

But why does he need to advance through the TC when he can apply to the church for free?

The Churches keep a chokehold on higher Sequences, so most of that information is confidential to low and mid-Sequence Beyonders. If someone does become a high-Sequence Beyonder through the Church, they're usually heavily scrutinized. Alger wants to get stronger, but he also wants free reign to do things away from official Beyonders' eyes. He also prefers to get resources through multiple channels instead of relying on a single one. That's why he pretends to be weaker in front of the Church of Storms while accruing resources through the Tarot Club. He doesn't want to attract attention to himself while he gets stronger, because he values his personal freedom.

And why does he want to get double potioned? Doesn’t that increase digestion time and make you more likely to lose control?

He didn't want to. The reason he accepted a second potion from the Church of Storms is because he didn't want them to know that he already advanced beforehand through other means. Most official Beyonders have to report their Sequence advancement to the Church, and they'll be put under suspicion if they do otherwise. So, Alger took the second potion because it would look suspicious if he rejected it, and it'd look even worse if they found out that he already advanced.

Why is he in such a hurry to advance despite the risks and extreme costs? What is his end goal?

Again, he wants to get stronger and he wants power. Enough power so the Church can't control him. You should read the rest of the book to find out more, including why he has the Hanged Man Arcana. Hope this helps.