There are guidelines for increasing the difficulty of an encounter. Basically, you calculate the XP budget for 6 characters. So instead of 100 XP for an easy encounter at L1, the XP budget is 150.
However I would not do anything at all with the encounters at first. Wait until your PCs are consistently breezing through them. Most groups need some time to get their feet under them, and the HotDQ part 1 encounters are not cake walks for parties of 4 PCs. Let them roll the encounters in part 1 if they can. Note you may want to use milestone leveling instead of actual XP for at least this level since they may not earn enough XP to advance this way.
That said, if you're looking for the actual guidance on leveling up the encounter, you'll need to deconstruct them a bit before you start. Take the monsters and recalculate the award XP, then figure out what you want to add. Use the chart from page 57 of the BD&D DM book to note what the difficulty multiplier for specific monster groups is (for instance if you have 2 monsters, the difficulty of their XP is multiplied by 1.5).
So let's take our encounter of 6 kobolds. Kobolds are worth 25 XP each that means our base encounter level is 150, however, with 6 of them that gives an encounter difficulty XP of 300. That's a medium encounter for 4 PCs. So if we want to up this by adding more kobolds, we'd need to change our difficulty modifier (since 7 is the next level). If we know our encounter budget is now 450, then we can divide that by 2.5, which gives us an actual budget of 180 XP, dividing this by our Kobold's 25 XP, we get 7.2 which tells us we need to add just one extra kobold.
This does leave us in a bit of a lurch when it comes to advancement XP. Since we add 50% more characters, but only ~15% more XP. This will be corrected by most encounters not needing this kind of scaling (use groups of different monsters more often than not). But can also easily be corrected by introducing the optional level up rules that ignore XP (what I'd recommend).
One other thing I've played around a bit with and haven't had enough time (or diverse enough groups) to do a full play test on is the idea of playing with the average HP and damage of the monsters. Basically, I always use the average HP and damage numbers for my monsters. This provides a different kind of difficulty tweak than adding extra monsters. Basically you could adjust the HP and damage up a bit (take 2/3 of the die instead of half, or more) and automatically make the encounter more deadly. I've mostly played with this the other way, reducing difficulty without changing numbers for a 3 person party. It worked pretty well, but may prove much more deadly if you raise the damage too much and find yourself oneshotting PCs.
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.
Best Answer
Adding Variation to encounters.
First: Terrain.: You should begin by considering terrain. When dealing with typical groups terrain isn't as large an issue because the tanks tend to control the battlespace. With casters, terrain becomes all important because they're going to want to utilize crowd control effects and force creatures to rough terrain in order to maximize their ability to pick them off from range.
To expand further on terrain to encourage people to really consider room layouts:
Things like pillars, furniture, drapes, windows, and whatever the place uses to manage temperature (fireplace, braziers, torches, etc) are all powerful additions to improvisation and challenge. The big thing is actually utilizing these features.
If the players, being new and all, don't use them, start having your monsters show them what they can do.
For example, have an orc kick over a brazier of coals to burn away some tangled vines that were cast for crowd control.
Have reinforcements smash in through the windows from the floor above.
If your casters are using a lot of ranged cantrips, get a couple of creatures together to start pushing a table in front of them to take the brunt of the damage (until it's destroyed).
There's a lot you can do, the limit is really up to you and your players.
Second: using CR as a guide: the CR ratings in the Monster Manual are a good indication of difficulty, but understand that the CR rating is meant for a balanced group of 4, and that your group of 3 casters (especially if they're glass cannons) is going to throw things off quickly.
For instance, they will have absolutely no problems with any single encounter because of their power level. The easiest way to make fights balanced for a nuke heavy party is to throw a couple of CR level fights at them in a row to see how many spells they need to use, and to see just how much is too much to throw at once.
When you have a feel for that, you can use what I call the fodder mobs.
In and of themselves, these aren't a big deal. However trying to kill them all before the get to your casters can be problematic. A caster burning his fireballs on the zombies no longer has them for use against 3 ogres in the next room.
Third: Overkill. With a caster heavy party, if you don't have any tanks, you're going to be severely limited when it comes to engaging them with the usual creatures such as orcs, ogres, trolls, goblins, lizardmen and other common creature types. You always have the option of showing them exactly why running no tanks is a terrible idea, but if you're more interested in keeping the story going with the characters they've made, then the tailoring is going to be 100% from your end. This leads me to my final point.
Fourth: DM mutation mode. If you're running a dungeon of undead and your casters take mobile as a feat so they can just kite the zombies everywhere, start introducing modified undead elites. One in every five skeletons comes up with the Charger feat, allowing it to dash and take an attack as a bonus action. Some zombies come up immune to fire. Others use nothing but the Dash action on their turn and explode when killed or when within 10 feet of a player in a 20 foot radius.
Basically, come up with new and interesting things on the fly to throw at your players in order to challenge them when things get to be routine.
In fact, you could have persistent undead that unless consecrated with holy water return from the pile of bones, ash or bodies that they lay in in 1D6 minutes, forever dogging the party until the casters come up with a way to barricade the enemies behind something impenetrable (like dropping the ceiling behind them somehow.)
Caveat on changing monster stats: you should always compensate in some other fashion in order to retain balance, and so you don't have to adjust XP or CR levels. For example, if you have a zombie come up with fire immunity, consider removing a HD from it's HP total in order to compensate the change. I personally prefer to leave this entirely up to random chance, and simply have tables of things I can add to my monster types in order to increase or decrease the complexity level accordingly. For instance, there's plenty of immunity or resistance to normal weapons that is bypassed by magical weapons, but sometimes it's a lot of fun to make a monster immune to any magic whatsoever and require the party to engage it using psionics, improvised weapons and regular weapons. Of course, since the monster is immune to magic, it doesn't have any magically enhanced strength or speed, so it doesn't hit like a freight train or use spell like abilities.
Anyways, try some of those out and see how they work for your group.