What really happened at the end of "The Talented Mr Ripley"?

1) How did Tom Ripley manage Meredith Logue kiss thing before Peter?

Earlier in the film, Ripley pretends to be Dickie and meets Meredith. He then really meets Dickie and acts as himself, before finally murdering Dickie and assuming both Dickie's identity and his own identity. Finally, as the police close in and look for Dickie, Ripley subtlety distances himself from the now-fictional identity and asserts his innocence.

All of this works, he is left with a substantial amount of Dickie's trust fund, and the police leave him alone.

This builds us up to the climax of the film.

There, Ripley goes on a cruise with Peter. However, to his dismay, Meredith is also on board. Remember, Peter thinks Ripley is just Ripley. But Meredith still thinks he is Dickie.

He realises that both Meredith and Peter can compromise his identity, as they both view him as a different person. This would obviously demonstrate his guilt in Dickie's disappearance and would be disastrous for him. When he meets Meredith on the cruise, the screenplay says:

Meredith points to the UPPER DECK BALCONY where TWO OLDER COUPLES are walking around the deck.

RIPLEY Of course. Aunt Joan.

MEREDITH And co. A lot of co. Oh, God, I've thought about you so much.

RIPLEY I've thought about you.

And now he's thinking I can't kill them all...

At the emboldened part shows, he can't kill her, as she has far too many family and he has no easy way to make this happen. The screen play shows he panics a little:

MEREDITH You know, the whole world thinks you killed Freddie? It's terrible.

RIPLEY I know. Look, I can't talk now. Later. Later?

He kisses her. Full of future.

MEREDITH So - are you travelling under R?

RIPLEY You know what - I am.

MEREDITH Dickie, are you with Peter Smith- Kingsley? I bet you are. My aunt thought she saw him.

RIPLEY (horrified) Peter Smith-Kingsley? I haven't seen him in months. No, I'm alone. (and he understands this is not any kind of lie)

So he kisses her to both settle her questions, impose a little control over the situation, and give him some space to consider how to resolve it.

2) Did Tom Kill peter also at the very end of movie?

Yes. In the very next scene, we see him strangle Peter to death. This is because, as he can't kill Meredith, he has to kill Peter. He sees it as his only option.

However, note the screenplay's ending:

The door of his closet flips open with the swell and he catches his reflection. It swings shut. Open then shut. Through the porthole the weather's changing as the light dies. There's a swell as the horizon rises and falls in the round glass. Ripley, alone, in a nightmare of his own making.

There's not an indication that this murder has suddenly solved his problems. It's very likely Meredith will at some point will connect that people think "Dickie" is actually Ripley and his disguise will be blown.

How he resolves this is left unclear. All we know at the climax of the film was that, whilst he thought he had gotten away with everything, he's been dragged right back into difficulty again.

I don't know if you noticed this, the motor launch that drove off with Mr. Greenleaf had "Andrea Doria" written on its stern, the Adrea Doria was an ocean Liner that sank in 1956 while bringing passengers back to USA.

If that was the ship that the movie ended with, the body of Mr Ripley's last victim would have gone down with the ship, and Ripley could be in action again, scott free.