The final challenge at the Pokemon League is against the Elite Four, a supergroup quartet of powerful trainers who use the game's most powerful types. Though each one has a "main" type, it's more accurate to say they specialize in two types, mostly ones that cover the bases of another. You'll need an incredible team to beat them; anything less than that (bar using an absurd amount of items) will only give you trouble. I recommend having a team at the high end of being Level 40s or early 50s when going in. You can't stop to heal at the Pokemon Center or buy more items at the Poke Mart so be fully prepared going in.
The single best Pokemon for this challenge is none other than Lapras. It will wall Lorelei's team and strike back with Thunderbolt, decimate Bruno with a mixture of Psychic and Surf, rip through Agatha's team with Psychic, and completely destroy Lance with Ice Beam. I highly endorse Lapras. Another great Pokemon is Gengar, who can use a variation on this set with Thunderbolt and Psychic and two other strong Special moves (or support). It has the added bonus of being invincible against all Normal-type and Fighting-type moves in the tournament.

Elite Member #1 - Lorelei

Elite Four: Lorelei
Specialty: Water & Ice-type

Details:
The first member of the Elite Four, Lorelei, has an excellent team to start off against. Although she says freezing moves are powerful, only two of her Pokemon can freeze you. Don't think you're off the hook though; when frozen in Generation I, you will never defrost unless hit by a Fire-type move, and unfortunately for you none of her Pokemon carry one. Her leading Pokemon, Dewgong, will use Rest to heal off all damage, so try and take it out quickly or during the turns it's asleep. The move Thunderbolt is excellent for that, and even moreso on a Lapras who can essentially wall her entire team. Slowbro needs to be dispatched extremely quickly as well as Amnesia is frightening: it boosts the Special stat by two. Cloyster has ludicrous Defense and will trap you with Clamp the way Blaine did with Fire Spin. Jynx isn't as bad as the rest of the team if you use strong Rock-type moves, but watch out for a mean Ice Punch from her should you break out a Golem or Onix. Her Lapras is also a terrifying with its Confuse Ray and Blizzard combo - your own Lapras would be the best counter.

Elite Lorelei DewgongCloysterSlowbroJynxLapras
Battle Type
Single Battle
Items
Level 54Level 53Level 54Level 56Level 56
Attacks:
Growl
Aurora Beam
Rest
Take Down
Attacks:
Supersonic
Clamp
Aurora Beam
Spike Cannon
Attacks:
Water Gun
Growl
Withdraw
Amnesia
Attacks:
Doubleslap
Ice Punch
Body Slam
Thrash
Attacks:
Body Slam
Confuse Ray
Hydro Pump
Blizard
Elite Member #2 - Bruno

Elite Four: Bruno
Specialty: Rock & Fighting-type

Details:
Undeniably the easiest trainer in this challenge, Bruno is still a force to be reckoned with. His team is very akin to Brock and the Fighting Dojo master's teams, using Onix as well as the Hitmons as main staples. His Onix share the exact same movepool; if you finish one off you'll have no problem with the other. Hitmonchan's moveset is practically useless save for Counter, which can totally destroy you if you're taking the route of powerful Flying-type moves on the Fighting-types. Hitmonlee has Jump Kick and Hi Jump Kick, both extremely powerful attacks. A recommended strategy is to use Fly on it with something; if it tries to Jump Kick or Hi Jump Kick you in the turn you're in the air, it will miss and take bad damage from it, setting up for a guaranteed KO the next turn with Fly. Machamp is the only true threat here as it has a great moveset which includes the OHKO move Giovanni gave you. If you have something impervious to ground you'll have no troubles though. A Gengar works wonders here as it avoids all Fighting/Flying moves and strikes back with a buffed-up Psychic.

Elite Bruno OnixHitmonchanHitmonleeOnixMachamp
Battle Type
Single Battle
Items
Level 53Level 55Level 55Level 56Level 58
Attacks:
Rock Throw
Rage
Slam
Harden
Attacks:
Ice Punch
Fire Punch
Thunderpunch
Counter
Attacks:
Jump Kick
Focus Energy
Hi Jump Kick
Mega Kick
Attacks:
Rock Throw
Rage
Slam
Harden
Attacks:
Leer
Focus Energy
Fissure
Submission
Elite Member #3 - Agatha

Elite Four: Agatha
Specialty: Poison & Ghost-type

Details:
This old bird makes great use of the Gastly lines, stuffing her team with two Gengars and a Haunter. This generation has an issue with the type match ups - because of that, Psychics are impervious to Ghost-type moves. A good Pokemon here would be Alakazam, who can sweep the entire team with just Psychic and will have the speed to beat out Gengar and Golbat. As long as you don't get hit by physical attacks you'll be golden. This team loves status moves, with everything being able to do almost anything under the sun, including poison, sleep, confusion, and paralysis. Don't afford them any of those luxuries - confusion will ruin a physical sweeper if it hits itself, sleep will leave you susceptible to Dream Eater (which most of the team save for the non-Ghosts runs), paralysis will lower your much needed Speed, and poison is, well, poison. Ground-type moves are great here too because the Gastly line doesn't have Levitate in this generation, leaving four fifths of the team open to a strong Earthquake or Dig. Don't bother with Normal or Fighting-type moves here though because they'll be useless on the Ghosts. If you are put asleep at any point, remember you can save healing items by using the Poke Flute to awaken.

Elite Agatha GengarGolbatHaunterArbokGengar
Battle Type
Single Battle
Items
Level 56Level 56Level 55Level 58Level 60
Attacks:
Confuse Ray
Night Shade
Hypnosis
Dream Eater
Attacks:
Supersonic
Confuse Ray
Wing Attack
Haze
Attacks:
Confuse Ray
Night Shade
Hypnosis
Dream Eater
Attacks:
Bite
Glare
Screech
Acid
Attacks:
Confuse Ray
Night Shade
Toxic
Dream Eater
Elite Member #4 - Lance

Elite Four: Lance
Specialty: Dragon-type

Details:
The king of the Elite Four, Lance, has the most frightening team of all. Gyarados, his lead, has decent Special in this generation and can wipe you out with Hydro Pump or Hyper Beam depending on what works better. Hyper Beam, by the way, is a staple of his team, and has an interesting quirk - if it knocks a Pokemon out it won't need to recharge. Luckily Gyarados has a double weakness to Electric-type moves, so hit it with those for maximum punishment. His two Dragonairs share a moveset that includes Dragon Rage (a set 40 damage move), Agility, Slam, and Hyper Beam. Dispose of them quickly or use a Rock-type with great Defense to sponge their moves and whittle them down. Rhydon or Golem works wonders here, as does an Omastar with Ice Beam. Aerodactyl is fast and has great Attack but poor moves - in fact, a Gengar can wall it completely. The main attraction, Dragonite, can set up a Barrier to boost Defense then take you out with a Hyper Beam backed by one of the highest Attack stats in the game. It does have a double Ice weakness though - exploit it.

Elite Lance GyaradosDragonairDragonairAerodactylDragonite
Battle Type
Single Battle
Items
Level 58Level 56Level 56Level 60Level 62
Attacks:
Hydro Pump
Dragon Rage
Leer
Hyper Beam
Attacks:
Agility
Slam
Dragon Rage
Hyper Beam
Attacks:
Agility
Slam
Dragon Rage
Hyper Beam
Attacks:
Supersonic
Take Down
Bite
Hyper Beam
Attacks:
Agility
Slam
Barrier
Hyper Beam
Champion - Blue

Elite Four: Rival
Specialty: None

Details:
The circle is now complete. Your first and last challenge is with your rival, and he'll have just as many tricks as ever to face you. With a fully rounded out team you'll be hard pressed to find one Pokemon to take everything out; however, if you don't mind legendaries, I recommend Zapdos for its Electric-type (good for beating literally up to 2/3 of the team depending on who he chose, or at the very least half) and Flying-type (provides STAB and good speed to take out Alakazam with a quick physical move). If not, balance everything out as always: don't train just one Pokemon (seriously don't) and make sure your type coverage is astute. His team has a few flaws you can exploit: Arcanine, if he has it, laughably knows Ember as its Fire-type attack, Gyarados's Hyper Beam makes it stall a turn if it doesn't wipe you out, Exeggutor only has three attacks (both damage-inflicting ones of the Normal-type), and Rhydon has a low-accuracy OHKO move it'll try on you. An alternate approach to this is to just use Toxic on everything (use PP Up if you're doing this) and stall it out with Chansey or another bulky Pokemon. It's not exactly enthralling but if it's what works go for it. Good luck!

Champion Blue Pidgeot Alakazam Rhydon Arcanine Gyarados Venusaur
Battle Type
Single Battle
Items
Level 61Level 59Level 61Level 63Level 61Level 65
Attacks:
Wing Attack
Mirror Move
Sky Attack
Whirlwind
Attacks:
Psybeam
Psychic
Reflect
Recover
Attacks:
Leer
Tail Whip
Fury Attack
Horn Drill
Attacks:
Roar
Leer
Ember
Take Down
Attacks:
Dragon Rage
Hydro Pump
Hyper Beam
Leer
Attacks:
Growth
Mega Drain
Razor Leaf
SolarBeam
Squirtle Chosen
Champion Blue Pidgeot Alakazam Rhydon Gyarados Exeggutor Charizard
Battle Type
Single Battle
Items
Level 61Level 59Level 61Level 63Level 61Level 65
Attacks:
Wing Attack
Mirror Move
Sky Attack
Whirlwind
Attacks:
Psybeam
Psychic
Reflect
Recover
Attacks:
Leer
Tail Whip
Fury Attack
Horn Drill
Attacks:
Dragon Rage
Hydro Pump
Hyper Beam
Leer
Attacks:
Hypnosis
Barrage
Stomp
Attacks:
Fire Blast
Rage
Slash
Fire Spin
Bulbasaur Chosen
Champion Blue Pidgeot Alakazam Rhydon Exeggutor Arcanine Blastoise
Battle Type
Single Battle
Items
Level 61Level 59Level 61Level 63Level 61Level 65
Attacks:
Wing Attack
Mirror Move
Sky Attack
Whirlwind
Attacks:
Psybeam
Psychic
Reflect
Recover
Attacks:
Leer
Tail Whip
Fury Attack
Horn Drill
Attacks:
Hypnosis
Barrage
Stomp
Attacks:
Roar
Leer
Ember
Take Down
Attacks:
Hydro Pump
Blizzard
Bite
Withdraw
Charmander Chosen