Chloe can function as a DPS with the usual setups—triple Skadi, or two Skadis and an Oberon, or even a Skadi/Castoria shell—but the majority of her value comes from her support potential. There are, broadly speaking, two ways to use Chloe as a support, which I’ll refer to as the “free damage” and “free battery” setups.
The free damage setup ignores the charge potential on Chloe’s third skill, swapping her out as soon as possible to maximize damage on all turns. The idea here is that you treat your team as you would any other farming setup, except you start with Chloe and use all three of her skills on turn 1. For a Buster setup, this will usually mean running Chloe/DPS/Koyan/Koyan/Oberon, with a 50% charge CE on the DPS. On turn 1, you use the DPS’s skills, plus Koyan 1’s damage skills. On turn 2, the second Koyan comes in, and you loop as normal. For Arts, you usually do Chloe/DPS/Castoria/Oberon/Castoria, with Black Grail on the DPS, and plug Castoria 1 for Castoria 2 on turn 1, using the rest of your batteries as needed. Quick skews closer to the Buster or Arts setups depending on the refund specs of the DPS in question.
The advantage of this is what it says on the tin: free damage. The Buster setup hits a little less hard than a conventional team on turn 1 but deals more damage on turns 2 and 3 than would otherwise be possible. For the Arts setup it’s pretty much pure benefit. This can make it easier to hit 90++ damage thresholds, which is always appreciated, though running a full set of SSR drop CEs is impossible. One thing that’s worth noting, though, is that the damage gain is such that a Buster Servant can run Dragon’s Meridian instead of an SSR charge CE and still come out ahead damage-wise. With Dragon’s Meridian on the DPS, Mash in the backline, and SSR CEs on everyone else, the team cost comes out to 113, which is feasible for high-level players. There are drawbacks to this setup—most notably that you can’t farm bond for anyone not actively involved in farming—but it might enable 5CE clears of certain 90++ nodes that were out of reach before.
The free charge setup is a touch more complex. The idea here is that you keep Chloe on the field through turn 2, saving her second and third skills. On turn 1, you use her first skill, you let your DPS generate Manuscript Complete stacks, and then you use skill 3 to cash those out for charge before Chloe finally leaves the field. This further delays your maximum damage—your turn 1 damage is even lower than in the free damage setup, and you don’t hit full buffs until turn 3—but it gives you somewhere between 10% and 30% extra battery you wouldn’t have otherwise. This gives a truly massive amount of extra flexibility for Arts and (to a lesser degree) Quick teams. Nearly every AoE Arts unit can loop when provided an extra 30% battery, and even with less than that, this is a substantial improvement to effective loop specs.
The most interesting use for this, though, comes in Buster teams. Any Buster unit with a 50% battery on a 6-turn or lower cooldown can now loop CE-less, provided wave 1 has 3 enemies. To do this, you run Chloe/DPS/Koyan/Koyan/Oberon. On turn 1, you use the DPS’s battery, along with Koyan’s. On turn 2, Chloe provides 30% charge and gets ready to leave, and you plug Koyan out for Oberon, who uses all of his batteries on the DPS to secure the loop. On turn 3, Chloe swaps herself out for Koyan, who uses her 50% battery, making the DPS’s battery available again, bringing you to 100% charge. This can technically be a 6-drop setup if the drop CEs are 3* or 4* in rarity, or a 5-drop setup with SSR CEs… but the main value of this is letting your DPS run the Black Grail for extra damage. In most cases, what this does is make 90++ damage thresholds more accessible to certain Servants. Losing two drop CEs relative to 90+ is not generally worth it, but if you need Black Grail or starting charge to clear the 90+ node anyway, or if you value the other benefits of 90++ (such as bond or higher drop rates on incidental mats) more than pure event farm efficiency, this can have some value.
Prior to now, 30% charge and 50% charge on Buster AoE units was mostly interchangeable. 50% chargers got to skip Mana Loading, or could use event CEs that gave charge without needing them to be MLB, but aside from that they mostly worked the same as 30% chargers. This gives 50% chargers a more meaningful farming advantage, even if a situational one, which is kinda neat. Kukulkan is the biggest beneficiary of this, as she was already mostly on-par with the highest-damage Buster units, and now she can also loop with BG under the right conditions.
Chloe’s CE choice only matters in CQ contexts, as for farming she’ll just take a drop CE in most cases. For hard fights with Chloe as a support, CEs with short-term benefit are a good pick. Ox King increases her damage push for Buster Servants specifically, and anything that provides a burst of stars can help fuel crits. You can also snag her bond CE, which gives a free bonus stack of Manuscript Complete when she arrives on the field. As a DPS, the Black Grail is the best pick for damage. You can also run something like Traces of Christmas for cases where you’re trying to stall or are worried about loop specs.
Chloe’s best use case is as a support, so supportive CCs are a nice choice. CCs that remove buffs from enemies can be good, and the ones that provide NP damage buffs to the party are great if they aren’t already in use elsewhere. Alternatively, you can give her the usual crit damage CCs—Chloe should flood the field with stars pretty often when used as a DPS, and she’s light on innate crit bonuses, which makes them appealing (even if they’ll be washed out with Skadi buffs usually).