My suggestion:
Mirror PC stats, abilities and numbers to give them a challenge. If you have a fighter, wizard and rogue group, and you want to put them up against monsters, you'll need to set them against something tanky, something ranged, and something that can hit hard and fast.
Alternately, you could design special monsters like a Colossal Scorpion. Give the torso the tanky stats, and make each pincer a rogue type damage dealer. In addition, the stinger could be considered limited ranged (30'/60'). This would give the scorpion 3 distinct parts that can be targetted, as well as a main body that will kill the whole thing.
You can apply this to any set of monster enemies as well. An Ogre, axe throwing orc, and orc shaman would be a good trio to set up against PC's if you balance their stats according to the PC stats.
The problem you're going to be facing with trying to balance this is something you've already pointed out, the CR ratings don't mesh with single fights. It will take a good amount of experimentation to get a feel for it, but if you start with PC stats and incorporate those into monsters, you should be fine using that as a baseline.
Numbers version of level 6 characters:
Fighter - 65 HP - 2 handed weapon
Wizard - 32 HP - evocation
Rogue - 40 HP - Whip, throwing knives
Creature battle:
Colossal Scorpion +7 to hit (Mutli-attack, this creature attacks with two claw attacks and a stinger attack)
Legendary Resistance 1 - Once per encounter, this creature can turn a failed save into a success
Regenerate - As a bonus action, the scorpion can sacrifice 40 Torso HP to fully regenerate a destroyed claw or stinger. That claw or stinger only becomes available for use on the creatures next turn.
Torso - 150 HP - damage resistant non-magical
Left Claw - 40 HP - Pincer attack 1d10 + 3
Right Claw - 40 HP - Crushing Grip 1d10 +3, auto-grapples
Stinger - 40 HP - Acidic spray, ranged 30'/60' 1d8 + 4 acid damage
If you kill the claws and stinger, the scorpion dies. If you kill the torso, the scorpion dies.
That's just one suggested battle to keep things interesting. The main point is that a battle like that enables a monster to take as many actions as your players can, and it doesn't destroy the creatures ability to remain a challenge.
If you break the enemy down into trios as well, consider giving them similar abilities to player characters. This will ensure the challenge rating is appropriate. Try to avoid making the mistake of jacking up their AC, to hit bonus, HP and resistances. Keep it consistent with what the PC's have to maintain the arena feel to the tournament.
First, ask if it is really a problem. Are the rest of the players bored watching a one-man show, or are they amused and having fun watching this part?
If it really is a problem....
Call for a Roll
Shopping shouldn't take all session, and if the player has had a bit of the spotlight already that session, then there is no problem with saying,
DM: Alright, we need to move things along. You're trying to persuade the shop owner to give you a discount on what? What is the end result you're hoping for?
Warlock: The stakes, the wooden box and few bottles of potions.
DM: Great. Roll for persuation, with advantage.
**Warlock:**16.
DM: You talk at length and convice him to give you a 20 gp discount. Do you take it or leave it?
Don't always go this route, but sometimes you just have to move on.
While X, what are you doing?
Purposefully shift attention to the other players. It allows him to have his moment, but not loose
Warlock: Your stunning array of woodcraft is beautifully carved. How much would it take to buy one of these fine pieces?
DM: Okay, so the warlock is going to try sweet talk a deal with the shopkeeper. While he's doing that, what are you doing, Fighter?
You'll see Matt Mercer of Geek and Sundry's Critical Roll use this a lot when the party is shopping
NPC Shift Spotlight
Or even have the NPC make the shift.
Warlock: Your stunning array of woodcraft is beautifully carved. How much would it take...
DM[as NPC]: Yes, yes, sell your flattery elsewhere silver tounged worm. [to Dwarf] Do you hail from the kingdom under the mountain, Yes? So far from home. What dwarven work treasures do you have to trade?
Skill/Perception Interactions
Or if you have a player with high passive perception, pass that player a note while they're talking about something they notice, while the Warlock tries his thing.
You notice blood on the hem of the store keeper's tunic, and the shopkeeper is trying to hide it by holding his cloak close around him. As you watch him, something else seems off. Is his accent different than the last time you spoke with him? Where his eyes always that shade of gold? Something is just... off.
They can then choose what to do with the information. Maybe they let the warlock keep going to see what happens, but now they get to enjoy an aspect of it the Warlock doesn't know about. Maybe they will interrupt the Warlock, and try to get the party out of there. Maybe they will try to confront the shopkeep in the middle of the Warlock's corruption/manipulation/persuasion tactics. How fun for all involved if it played out like:
Warlock: Your stunning array of [DM slips not to Rogue] woodcraft is beautifully carved. How much would it take to buy one of these fine pieces? I'm sure I could pay a fair price of...
Rogue: I stab the shopkeeper while he's distracted.
DM: Roll for attack.
Warlock: Wait! What?
Best Answer
Nothing. You do nothing. A single lich will kill your party and eat their souls for breakfast.
It isn't possible. As a 3rd level character, there is absolutely nothing you can do to defeat a lich, barring your DM intervening with some Deus Ex Machina.
A lich has a challenge rating of 21. Based on my napkin math using the experience guidance in the DMG, three liches is a deadly encounter for ten 20th level characters. A single lich is a deadly encounter for two 20th level characters. A CR 18 demilich, which is a lich whose soul has decayed because of malnourishment, is a deadly encounter for two 20th level characters.
Hopefully, your DM knows something you do not.
Hopefully, your DM doesn't actually intend you to fight three liches, as it would be a total party kill in the first round. It seems quite likely that there is something else at work here. Philipp covers several conceivable possibilities in their answer.
If you aren't having fun, talk to the DM outside of the game.
Maybe your DM really is throwing impossible encounters at your party. This doesn't sound like fun, so if this is the case, talk to your DM about this. Being polite and respectful, just tell them, "I'm not having fun dealing with these impossible encounters. Can we get something more on our level?" And see how it goes. If your DM insists on messing with you, you may have to find a new game.
In universe, at least one of your characters should be able to determine the old man is blowing smoke.
You stated:
Surely you guys have heard of liches. Surely your characters would be aware that these are significantly powerful creatures. If in my game, we encountered an NPC asking is to dispose of 3 liches, we would chuckle and move on to the next quest giver. Your characters should probably know that this is impossible. You characters should know they shouldn't even attempt this.