Another option for a decent sword slinging DX based warlock is to start out as a rogue to get the proficiency in Short Sword/Scimitar, which is both light and finesse, which you will need for Sneak Attack from Rogue and to allow two-weapon fighting at LVL1 without a feat. Dagger also qualifies, but only does 1D4, but it can be thrown, so have one available!
Race could be Elf, if you want to start as Warlock as they have Short Sword proficiency from the race and is not a bad choice with +2 to DX, but any other race needs to start as Rogue to get the Short Sword proficiency needed and Half-elf is a great choice with the +2 to CH and +1 to DX and CN are the obvious other choices for Rogue/Warlocks.
RAW (Rules As Written) state that any character can have two attacks with two light weapons using their bonus action. (see Light weapon description PHB p. 148 and this from PHB p. 196) "When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative. If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.
Bonus actions are not something you need to earn or gain via a class or ability, the limitation is simply this. You have only one bonus action per turn and can only do things with the flag of "bonus action". Anyone could use a bonus action if they had an ability, spell, etc that must be used as a bonus action. Here's the Dice: SS = Short Sword 1D6, SA = Sneak Attack 1D6 at Rogue Lvl1, Let's assume Dex is moderately high 16 or 17 +3 to hit and damage with Finesse weapons.
So, starting off as a Rogue, LVL1, you get yourself two short shorts, each with 1D6 damage. The first attack gets, if qualified, the Sneak damage and Dex bonus to damage, second attack does not get Dx damage bonus. Assuming you hit both times, you just did 3D6 +3 damage or average 14 HP. Not bad for a 1st Lvl Rogue. But now things are going to get interesting as you multi-class into Warlock and take Hex as one of our two first Lvl Spells. Hex will add addition 1D6 to EACH attack you hit on, so that same scenario of hitting twice with your two short short while Hex is in effect on the creature(s) you attack your damage just became 5D6+3 ((SS1 + SA+ HEX +DX) +SS2 + HEX) or 21HP average. Now you can either continue with Rogue and gain an additional 1D6 every time you hit an odd numbered level or since this is about Pact of the Blade, continuing with Warlock to Level 3, my preference is to go with the Fiend for the synergy of Dark One's Blessing of tmp HP upon slaying foes, but other patrons still work great for Blade, Fey if your not an elf is a great choice.
Assuming we stayed at Rogue Lvl1 and when straight to on through Warlock Lvls bumping our DX up to 18 (+4) at Warlock4, and take the Eldritch Invocation of Thirsty Blade Lvl5 we now get another attack. This is where it gets a little tricky, but conservatively, this would work. To have the two main attacks and the bonus attack, RAW state that we must have light weapons, so we will make our Pact Blade a Short Sword. Why choose a weapon that only does 1D6 when we could use a Great Sword for 2D6? Because we are still trying to take advantage of our Rogue Sneak attack. You may be able to argue for a Rapier and go to 1D8 for your pact weapon, but it really doesn't change the coming numbers much. All those D6's really start to add up!
So, here's the numbers at Rogue1/Warlock5: SS1 + SA + Hex + DX + SS2 + HEX +DX + SS3 + HEX or 7D6 + 8. That's 32.5 Average. Now, we continue to increase Rogue to squeeze another D6 each couple of Levels. At Rogue3/Warlock5, it's 8D8 +8 now with 2D6 for our SA and if we choose to be an assassin, that first attack is an auto crit on surprise with SS + HEX + SA +DX or 8D6+4 for the assassin blow!
At Rogue4/Warlock5, we can bump DEX again to 20, to make our three attacks total 8D6 +10! At rogue5/Warlock5, SA goes to 3D6 and all three attacks with hits now total 9D6 +10. Now, granted, we are not always going to be able to use our bonus action as a third attack. That drops 2d6 for the numbers above when you need to use your bonus action for something other than the 2nd attack prior to Warlock5 or third attack thereafter.
All the while, if we also started with a decent Charisma of 16 or more, we have also been able to deal damage at a distance with Eldritch Blast for 1d10 at Lvl1 and get to add our CH bonus if we take Agonizing Blast Invocation at Lvl2 for 1d10+3. At Warlock5, it jumps to 2d10+6 (separate attack roles for each beam) for average of 14HP.
The big downside of this build is that lack of good armor and without a decent CON, you will not be able to maintain concentration on HEX or last in the melee. However, there are many other synergies with Fiendish Vigor invocation, Armor of Agathys, and if your DM allows feats(optional), forgo a DX bump for a CN bump, or go with Moderately Armored feat and/or Dual Wielder feat which allows you to go to two rapiers or a rapier for your SA pact weapon and another non-light weapon in the offhand and gain +1 to AC.
Also, continuing on in Rogue gives you Stat bumps at 4,8,10,12, along with the two others for Warlock at 4 and 8, so you'd get 6 stat bumps and/or feats if allowed by your DM. Which, by the way is the only way you get them. A lot of folks post that they would get a bump at Rogue1/Warlock3 (character LVL4), but that is NOT how it works in the RAW. Also, you only get the new proficiencies of the new class as shown on PHB p.164-164. You do total your levels for proficiency bonuses at character lvl 5, 9, etc and for Cantrip Damage.
All the while, you still have all the spells and abilities of a warlock along the way and all the cool stuff you get as a rogue, like the cunning action, expertise in skills, uncanny dodge and thieve's picks and tools, making you a very versatile character.