TFT Comp 1 - Duelists/Mystic/Adept/Exile
Units: YASUO, Fiora, JAX, Janna, KALISTA, Shen, Lee Sin, Yone
This TFT comp has been very competitive for quite some time now. You can try to force this composition even without having a Chosen Duelist (you are going to need a Spatula and a Bow for that), but of course, getting one would make things much easier for you. You want to start spending your excess Gold at level 5 as you are looking to get 3-Star Yasuo (1 cost unit) who's one of the main carries of the team along with Kalista. Also, if you can make Fiora 3-Stars in the process that would obviously be useful.
Once you've done that, it's time to start investing in levels. At level 7 you should start rolling again for a 3-Star Jax (2-cost unit) and at least a 2-star Janna. At level 8, or earlier if you can find him of course, Yone will give a huge boost to your Yasuo through the Exiled buff and will also give you the Adept bonus.
TFT Comp 2 - Warlords
Units: Nidalee/Garen, Jarvan, Pyke, KATARINA, Senjuani/Aatrox, TRYNDAMERE, Azir
This is another team composition that has been strong for quite some time. Katarina is the main carry here and the only real difference is the addition of Tryndamere (4-cost unit). You want to go for this team when you get a Chosen Warlord and you have the best items to give to Katarina (Hextech Gunblade + Quicksilver).
Since your carries in this one will be Katarina (3-cost unit) and Tryndamere it's important to keep a good economy and focus on leveling. Zed can be added to the team for the 3/6 Slayer bonus.
TFT Comp 3 - Chosen Diana
Units: DIANA, Yuumi, Kindred, Teemo, Katarina, AKALI, Talon, Pyke
New patch, but still good, old Diana. Even without the Moonlight trait, she can still be a solid carry. There is more than one way to build around her in this patch and it really comes down to the Chosen trait (Spirit or Assassin). You want to slow-roll your excess Gold at level 5, as it's very important to make her 3-Star before you start leveling up.
You are looking for 6 Assassins and 4 Spirits as your end-game composition. Talon or Akali can be your second carry on this team. Depending on the Chosen trait you will have to make an adjustment leaving out either 1 Spirit or 1 Assassin.
Units: MORGANA, Nasus, Swain, Vladimir, Irelia, JAX, Wukong, Kayle, Lee Sin
Morgana is the main reason why this TFT comp is so strong. Her ability now reduces Armor, Magic Resistance, and Attack Damage but it no longer heals her. This is where 4 Syphoners come into play, since their trait offers healing for 100% of their damage. (Morgana used to heal for 30% of her damage in set 4).
Even a 2-Star Morgana can be a very solid carry at the moment. You are looking for 6 Enlightened + 4 Syphoners as your end game composition. It's important to keep a good economy as it's much easier to make her 2-star at level 8. You will have to make an adjustment depending on the Chosen trait, leaving out either 1 Enlightened or 1 Syphoner.