Well, you've already gotten a decent answer about how the rage mechanics work with spells This answer is focused on how you might multiclass. Watch out, this is gonna be a bit long.
No matter what you decide to multiclass as, take Chill Touch as a cantrip, even if it's one you take later. I know Eldritch Blast looks like the best offense cantrip, but Chill Touch has an often overlooked ability:
On a hit, the target... can't regain hit points until the start of your next
turn. Until then, the hand clings to the target. (PHB, 221)
That is clutch if you're up against anything that can regenerate, or if the enemy group has a tank and a healer.
Warlock general advice
So first off, lets go over advice for your Warlock in general, regardless of the other class you take. I would say that you take the Fiend patron. That first level ability is going to be great for you
Dark One's Blessing: Starting at 1st level, when you reduce a hostile creature
to 0 hit points, you gain temporary hit points equal to your Charisma modifier + your warlock level (minimum of 1). (PHB, 109)
This is like Armor of Agathys on repeat, and it increases by 1 with every warlock level. I would also suggest the Fiendish Vigor invocation
Fiendish Vigor: you can cast false life on yourself at will as a
1st-level spell, without expending a spell slot or material components. (PHB, 111)
False Life: (Duration: 1 hour) Bolstering yourself with a necromantic facsimile of life, you gain 1d4 + 4 temporary hit points for the duration. (PHB, 239)
You can cast it on repeat and always have 5-8 temporary HP at the start of a battle, which can help out a lot if you're going to be a melee fighter.
When you reach 4th level in Warlock, I would suggest taking the feat Tough:
Tough: Your hit point maximum increases by an amount equal to twice your level when you gain this feat. Whenever you gain a level thereafter, your hit point maximum increases by an additional 2 hit points. (PHB, 170)
Which, at bare minimum, if you're multiclassing, would instantly give you 10 HP (4 levels in Warlock, 1 level in other), and then you'd basically go from +3 HP per level to +5 HP per level
The Barbarian Multiclass
If you really want to use Barbarian, one level is okay, but it might take a few more levels to really get how you want it; I'm going to suggest you take at least 3 levels in Barbarian, possibly 5 (if you want extra attack).
First of all, let's talk about Rage and Spellcasting.
- At first it seems like a big turn off, but you should remember that you only get 2 spell slots per short rest, and the Warlock actually has a decent selection of spells that can be used outside of combat. Besides that, the rage is fine to use after you cast a spell or two.
- Witch Bolt the enemy tank for the first round or two, and then when he closes the distance on you, rage all over his face.
With the Fiend, you can cast Command and order the target to Grovel, and then on their round they have to drop prone, which gives you advantage on melee attacks.
Or, the second level spell Blindness/Deafness is great because it's not concentration, so you can blind the enemy and then enter melee combat while they have disadvantage to attack.
To start, depending on how your hit dice roll, you may want to consider using a shield and a one handed weapon. It'll increase your AC to 16, and 11-13 damage is still decent for a weapon. You could always take it off and use a two-hander as well. I know you're thinking "I'll swap them with my pact blade", but that's actually less efficient
Pact of the Blade: you can use your action to create a pact weapon in your
empty hand. (PHB, 107)
Other Activity on Your Turn: ...You can also interact with one object or feature of the environment for free, during either your move or your
action. For example... you could draw your weapon as part of the same action you use to attack. (PHB, 190)
Thus you would have to spend one turn taking off the shield, a second summoning the weapon, and then attack on the third, as opposed to drawing a standard weapon and attacking with it on the second turn. To be safe, I would rage on the turn that you draw the weapon because
Rage: Your rage lasts for 1 minute. It ends early if you are knocked unconscious or if your turn ends and you haven't attacked a hostile creature since your last turn or taken damage since then. (PHB, 48)
But if you're up against a guy that you're pretty sure is gonna hit you before your next turn (like all of his attacks have been in the high 'teens to low twenties), I'd say risk it.
Also, I would suggest you get some throwing axes. First-thought logic would say "why bother, I have ranged cantrips?" These throwing axes are for when you're raging. You need to either attack someone or take damage, or else your rage drops. Thus, if someone else kills the guy you were fighting, and everyone else is out of your movement range, you can start moving towards someone and then chuck an axe at them, because the wording never says you have to successfully hit them to keep your rage.
Multiclassing continued
Alright, now on to why I suggest multiple levels in barbarian. So the second level gets you another d12 hit dice, and
Reckless Attack: ...When you make your first attack on your turn, you can decide to attack recklessly. Doing so gives you advantage on melee weapon attack rolls using Strength during this turn, but attack rolls against you have advantage until your next turn. (PHB, 48)
Danger Sense: ...You have advantage on Dexterity saving throws
against effects that you can see, such as traps and spells. To gain this benefit, you can't be blinded, deafened, or incapacitated. (PHB, 48)
These are actually pretty useful. If you're in a fight against a big baddy, who's just wailing on you every turn, with big attack bonuses putting them in the twenties every hit, and he's got super high AC, the advantage you get will outweigh the one he gets, unless he's got some special ability when he's got advantage, like a sneak attack. As for Danger Sense, I mean, who doesn't want advantage on Dex Saves versus spells and traps? Might save you from a fireball. Speaking of fire, let's go on to level three of barbarian. This is where you take your barbarian path. So take path of the Totem Warrior, Bear totem, because:
Bear: While raging, you have resistance to all damage except psychic damage. (PHB, 50)
Your damage resistance goes from non-magic piercing, slashing, and bludgeoning, to everything but psychic, which is pretty awesome in a fight, and you also get a third use of Rage. To top it off, once you reach 10th level in Warlock:
Fiendish Resilience: ...you can choose one damage type when you finish a short or long rest. You gain resistance to that damage type until you choose a different one with this feature. (PHB, 109)
You can choose Psychic damage, and suddenly your Rage makes you resistant to everything. Also, going to 3rd level in barbarian gives you three d12 hit dice to spend healing yourself during a Short Rest. If you go up to a 4th barbarian level, you get a Feat, which you could take as Great Weapon Master:
Great Weapon Master: On your turn, when you score a critical hit with a melee weapon or reduce a creature to O hit points with one, you can make one melee weapon attack as a bonus action. Before you make a melee attack with a heavy weapon that you are proficient with, you can choose to take a -5 penalty to the attack roll. If the attack hits, you add +10 to the attack's damage. (PHB, 167)
This one can turn you into a real heavy hitter, once your attack bonuses are high enough that the -5 won't do much.
Now, you can intermingle these barbarian levels in between several levels of Warlock, and you'll probably want to wait until after you've gotten your blade pact and 5 levels in Warlock, to take Thirsting Blade, for a double attack with your pact weapon.
Why Fighter might be a good choice
However, if you've been turned off from Barbarian, Fighter is a decent second choice.
The Fighting style Great Weapon Fighting looks tempting, but with the smaller hit dice I would really suggest you take the Defense style, as you gain a +1 to AC while wearing armor, which means you can start with the cheap Ring Mail at 15 AC and then after a few fights you can afford Chain Mail for 17 AC, and then eventually work your way up to the expensive Plate Mail at 19 AC, and you can take the feat Heavy Armor Mastery:
Heavy Armor Mastery: Increase your Strength score by 1, to a maximum of 20. While you are wearing heavy armor, bludgeoning, piercing, and slashing damage that you take from nonmagical weapons is reduced by 3. (PHB, 167)
Just keep in mind that Heavy Armor takes 10 minutes to put on (PHB, 146), so if you're not wearing it and get ambushed, you're stuck with 11 AC for the fight (13 if you've got a shield), so you'd probably want to have Mage Armor prepared, which takes up one of your known spells.
Second Wind is also nice, giving yourself 1d10 + 1 hp. You might also want to take extra levels in it, as Action Surge can be useful, and if you get to 3rd level, you can take Champion and critically hit on a natural 19 as well as a natural 20.
Paladin as an option
Paladin might also work for you, if you're going to take more than one level.
I Would still suggest Defense fighting style. Divine Smite would mix with the Pact of the Blade well, and you'd have Lay on Hands and a few smite and healing spells. If you're willing to delve more into the Paladin behavior, you could probably do the Oath of Vengeance well, and Vow of Enmity is good for melee fighters:
Vow of Enmity: As a bonus action, you can utter a vow of enmity against a creature you can see within 10 feet of you, using your Channel Divinity. You gain advantage on attack rolls against the creature for 1 minute or until it drops to O hit points or falls unconscious. (PHB, 88)
Combine that with how the Divine Smite says "when you hit a creature with a
melee weapon attack," (PHB, 85) and you can wait until you get a critical hit.
