Now, to start, let's talk about how to bring Percy's power up to the standard of the DC Universe. Not to say that he is weak, just not on the level of Wonder Woman or Superman. He is surprisingly strong, being able to hold up the sky with Annabeth's help, but he gains even more strength when in water, proportional to a God. And that's what I'm going to use as a method to increase his power. When he crosses over, his godly blood will begin to draw new power from the DC Universe's Greek Gods. His demigod powers will grow in strength to match this new source and he will now be a match for, say, Diana or Aquaman. This new connection would also give him access to the Clear, which gives Aquaman his potent telepathy.
So, since we've established how he will get stronger, but now let's talk about his actual original powers. Obviously, we'll start with the basic demigod abilities: It's your standard affair of enhanced human abilities. Strength, speed, reflexes, durability, you name it, it's probably far above the average man. Of course, he's much more powerful than a demigod born to say Aphrodite, since his father is one of the Big Three. So, that's the original Percy, the new Percy would have at least triple that amount of power. Now on to his power inherited from his father. Most of these are branched off from water or rather his hydrokinesis. From this ability, he gains limited forms of cryokinesis, toxikinesis, and vitakinesis. He can freeze water in the air into ice and snow to create small blizzards, control and manipulate poisons, as well as cure wounds and poisons when in contact with water. His is also completely immune to water-based attacks, and also has a very high degree of heat resistance.
Now for the powers that stem from his atmokinesis, which isn't nearly as powerful as a son of Zeus. He is able to create storms, such as hurricanes and thunderstorms, which are extremely powerful, as they are able to extinguish the fires of Hyperion. Through this, he gains a limited form of electrokinesis. He also gains some aerokinesis, considering his apparent ability to summon strong winds. And finally, his last true power is geokinesis, which is displayed in his ability to cause earthquakes and volcanic eruptions. Included in his repertoire of abilities gained from Poseidon is his ability to speak to both aquatic and equestrian life.
Moving on to how Perseus gets to the DC Universe, we'll need to discuss possible outcomes for his life. I want him to not have any baggage when he arrives to New Earth, so he'll need to have lived a full life i.e. had kids and grand-kids, dying when he reached the end of his natural lifespan. Annabeth would have passed before him, due to his being Poseidon's child. In his final moments, Chaos comes to him and offers to take him to a new world. Curiosity taking hold, Percy decides to take the offer, with Chaos removing his soul from his body. He then takes him to meet the Presence, and both begin to rebuild Perseus' body, deciding on a 14-year old appearance. Chaos also brings along Riptide, enhancing it to the point of indestructibility, but the Presence also has a gift: Blackjack, as a foal.
Now, after all that, where does Percy end up after his arrival? Well, I would propose that he awakens on the Island of Themyscira, where the various Amazons would interrogate him, and he would meet Diana, who at this point is still 13. He'd explain his situation, i.e. his heritage, and be raised and "trained". As for romance, you can already tell who I've chosen: Wonder Woman. Of course there are other choices for love interests, I just felt that Diana would be the most appropriate.