Jynx is a quick Pokemon that can surprise people if played properly, and is one of the strongest Ice-type attackers in the game due to its decent Special and STAB Blizzard, but ultimately it is fairly fragile. You'll probably want to lead with Jynx, as you can't switch it into much of anything but Psychic, Ice Beam, or Blizzard. Keep your Jynx far away from Normal Pokemon, Starmie, and Slowbro and she will prove her effectiveness. This is a Pokemon that you have to practice with more than other standards to properly execute.
If Jynx is on your team, it is best to use it as your lead Pokemon. Jynx is the fastest Pokemon in the game with a sleep move with at least 75% accuracy, and holds an advantage over the most common sleep lead, Exeggutor, due to Exeggutor's Ice weakness and lack of Speed. Jynx's downside is that it has poor defenses and is slower than Tauros and Persian, both of which threaten it.
You do not necessarily want to use Lovely Kiss on your opponent's lead, because they might switch to something that is less useful to put to sleep, and if you can predict well, then you want to put to sleep the biggest threat you possibly can. If you are not comfortable making that prediction, then just use Lovely Kiss on the first turn, because putting something to sleep is almost always better than putting nothing to sleep. A likely switch-in after you have put something to sleep is Starmie, and you can make great use of three of Starmie's four likely attacks if you use Mimic—Thunderbolt, Thunder Wave, and Recover.
Psychic from Jynx prevents, among other things, a Pokemon such as Gengar from safely switching in and putting Jynx to sleep. However, remember that if you keep Jynx in against a lead Gengar, there is a 60% probability your Jynx will be put to sleep; the only way Jynx will OHKO Gengar is with a critical hit, although Jynx has a high enough Speed and therefore a fairly good chance for a critical hit. Jynx will, in most situations, do the most damage with Blizzard, so there is not often a good reason to use Psychic. The only reason to use Psychic is if you are in a situation that you cannot afford a miss or if you want to cause a Special fall.
Body Slam can catch Starmie on the switch with paralysis, but do not expect to do much damage with it. The chance of paralysis that Body Slam offers might often be more useful than anything you can Mimic, so this is a very strong option to consider. Counter can KO Tauros when it does not critical hit, and although Tauros can KO Jynx with a non-critical hit Hyper Beam (the damage range is 85.9%-101.2%) Jynx has a good chance to survive and counter if at full health. Counter is an underrated option on Jynx, but it is only useful in a limited amount of situations. Snorlax is going to Body Slam Jynx, and if it already switched into a Blizzard for 30-33% damage, then Counter is going to most likely KO it if it Body Slams Jynx. Minimum damage from Counter on the same turn as a Tauros Body Slam will do 93% damage to Tauros. However, if either Tauros or Snorlax were to critical hit, they would KO Jynx, and if it were to be paralyzed, you might lose Jynx without doing any damage; also, it will be crippled for another Pokemon to come in and finish it off. Tauros and Snorlax can also just attack Jynx with Earthquake if they are really scared of the possibility of Counter.
Rest is also a good option, which Jynx can get off against an unsuspecting Chansey, which cannot do anything significant to damage it besides Seismic Toss, or Exeggutor, which can only Psychic or Double-Edge Jynx and is forced out when Jynx wakes up. It is highly unlikely that Exeggutor would use Explosion on Jynx, but it is also unlikely that it will stay in and let Jynx Blizzard it, so expect Exeggutor to switch out.
Checks and Counters
Starmie can paralyze Jynx, resists Blizzard and Psychic, and can do neutral damage to Jynx with Thunderbolt. However, if your Jynx manages to Mimic Thunder Wave or Thunderbolt when Starmie switches in, you might reconsider staying in. Alakazam, likewise, can paralyze Jynx, and resists Psychic if it manages to switch in on that. It does take somewhat significant damage from Blizzard, but it still gives Jynx problems. Slowbro can switch in, get Amnesia up, and resists both Blizzard and Psychic. Even if Jynx manages to Mimic Amnesia, Slowbro completely destroys it due to type match-ups. Chansey, although it hates the chance of being frozen, takes almost no damage from Jynx's moves, while doing plenty of damage to any Jynx without Rest. Tauros and Snorlax will destroy any Jynx without Counter, and will certainly OHKO Jynx with a critical hit.