Thundurus as a check on Flying/Electric
Thundurus is immune to Klefki's Thunder Wave so there's not much Klefki can do to cripple Thundurus. Thundurus has access to Prankster as well as Taunt which could potentially shut down Klefki. The reason it's a check and not a counter though is because Taunt Thundurus is situational just like Klefki not having Foul Play, which is also situational. Klefki's Foul Play 3HKO's Thundurus but Thunderbolt 3HKO's Klefki even under Light Screen (that's assuming it's holding Light Clay and not Leftovers) so Thundurus kills Klefki before Klefki kill it. Klefki also needs a turn to use Light Screen so Thundurus definitely beats it 1v1 if it's the standard Klefki spread.
252 SpA Life Orb Thundurus Thunderbolt vs. 252 HP / 0 SpD Klefki through Light Screen: 104-123 (32.7 - 38.6%) -- 99.5% chance to 3HKO
If it's SpD Klefki however than it can beat Thundurus 1v1 with Foul Play + Life Orb recoil.
Mandibuzz as a counter on Dark/Flying
The only utility Mandibuzz provides in the match-up vs Fairy is Defog which it can only do vs Klefki that gets rid of Klefki's screens on top of hazards, so even if it has to take a Thunder Wave it won't matter too much as there's nothing else Klefki can do to it and is forced to switch out.
Magnezone as a counter on Steel/Electric
Magnezone is immune to Thunder Wave, resists both of Klefki's STABs if it's running them and doesn't take much from Foul Play as it is a special attacker. It also gets access to Magnet Pull which prevents Klefki from switching out and can proceed to kill it off with Thunderbolt or Flash Cannon.
Garchomp as a counter on Dragon.
Garchomp is immune to Thunder Wave and can beat Klefki with Earthquake even if it wants to set-up a Reflect. If the Klefki has Magnet Rise Fire Blast will also get the job done as Klefki can't touch Garchomp.
Mega Diancie as a check on Fairy/Rock.
Mega Diancie's Magic Bounce ability prevents Klefki from spreading paralysis and setting up Spikes. Even with Light Screen up Mega Diancie is still gonna be able to beat it with Earth Power or Hidden Power Fire. The reason it's listed a check is because Klefki has the potential to run Flash Cannon even if it's not commonly and there are better options over it.
Galvantula as a check to Klefki on Bug.
Galvantula is immune to Thunder Wave which means Klefki can not cripple it. Thunder under Light Screen to a physically defensive Klefki does 25.7 to 30.5% which means it's a guaranteed 4HKO. If it's a specially defensive variant of Klefki it's gonna be a 5HKO at least and Foul Play from Klefki can 4HKO it with Foul Play.
252 SpA Galvantula Thunder vs. 252 HP / 0 SpD Klefki through Light Screen: 82-97 (25.7 - 30.5%) -- guaranteed 4HKO
252 SpA Galvantula Thunder vs. 252 HP / 252+ SpD Klefki through Light Screen: 58-69 (18.2 - 21.6%) -- possible 5HKO
0- Atk Klefki Foul Play vs. 0 HP / 0 Def Galvantula: 75-89 (26.6 - 31.6%) -- guaranteed 4HKO
Gliscor as a counter on Ground.
Gliscor can't be touched by any of Klefki's moves and can proceed to kill it off by using Earthquake.
Nidoking as a counter on Poison.
Nidoking can't be paralyzed by Klefki's Thunder Wave and can easily beat it with Earth Power.
Mamoswine as a counter on Ice.
Mamoswine being a Ground/Ice type can't be crippled by Klefki and beats it with Earthquake.
Diggersby as a counter on Normal.
Diggersby is immune to Thunder Wave which means Klefki can't do much more it as it's only job is supporting the rest of the team. Diggersby can finish it off with Earthquake.
Swampert as a counter on Water.
Swampert among other Water/Ground types counter Klefki as they can kill it off with Earthquake and not be touched by Thunder Wave.
Infernape as a counter on Fighting.
Not much that Klefki can do to Infernape other than paralyse it. But it's gonna get beaten by Fire type attacks.
Victini as a counter on Psychic.
Victini being part Fire type can take on Klefki even through paralysis. It's listed as a counter assuming there's no luck involved because if Victini gets paralyzed twice in a row Foul Play can 2HKO it.
Landorus as a counter on Flying.
It's immune to Thunder Wave and Klefki doesn't have much to touch with, but Landorus can take it out with a Ground type move.
Chandelure as a check on Ghost.
Although it has to take a Thunder Wave it's Fire typing allows it to take on Klefki with Fire Blast or Flamethrower.
Mega Sableye as a counter on Ghost/Dark.
Mega Sableye prevents Klefki from spreading paralysis and setting up Spikes. It still allows it to set up screens but other than that it forces it to switch out and it can wear it down with Will'O'Wisp as Klefki doesn't have anything to touch it.