Vladimir's laning phase is "weak" in the sense that he doesn't do much damage in a burst pattern, meaning if you get into his range, you can deal quite a bit of havoc to him.
There are many major roadblocks to dealing actually significant damage to him, and I will list them right now.
1. His Q's range and healing. Thanks to having a ranged heal that also functions as a cs helper / harassing ability, it's hard to make meaningful trades with Vladimir as he can heal back up and deal damage to you at the same time.
2. Fizz E on steroids. Vladimir goes untargetable for 2 whole seconds at the cost of 10% of his CURRENT health. It gets worse, as he's briefly hasted 37.5% which exponentially decays for 1 second, so he's usually faster than you as well during this duration. This also makes ganking him extremely difficult.
3. Phase Rush, and his lack of necessity for damage. Bonus movement speed on a low cooldown allows for him to make extremely fast trades, as he can tap E to gain the third stack for Phase Rush, and then run off before you can engage succesfully.
"But if his laning phase is so safe, that means that he won't be able to make too much use of that mobility in the trading patterns right?"
It literally doesn't matter because Vladimir brings TWO things to the table. That movement speed still helps him out in achieving these two things though.
1. Teamwide burst
2. Incredible healing.
With 2 items Vladimir begins to hyperscale thanks to his passive being Ryze's passive but Health. With 2 items he can burst squishies. It only gets worse after that.
"What about grevious wounds? Executioner is only 800 gold, right?"
An 800 free gold advantage for just picking Vlad by melee champions that likely aren't going to be able to proc it anyway?
Not to mention it doesn't even matter if you're playing someone like Darius. Vladimir has a built in untargetable in his kit, and can simply use W and pool out of Darius's range thanks to the brief haste it grants, and Vlad can also utilize Phase Rush if not even that is enough. There's a reason Vladimir doesn't have to run ANY damage in his runes, and that reason is that he can just play safe until he can delete an enemy team. Sure, building Executioner on Vayne might help for late game because then you can build Mortal Reminder... and that's it. If you're playing AP into Vlad you can build Morello, which you will likely do later anyway, but it'll take you 3000 gold's worth of time to POTENTIALLY get it if you're not extremely mana dependent or item dependent. (Nearly no one)
Vladimir does so well, so who can counter him?
People with even faster trading combos, like Quinn, Yorick, or Kled, can burst him / engage on him before he can get away. Yorick can use his ghouls and hardshove lane while healing and ignoring Vlad's harass, Quinn can burst him down faster through E and Q (and can build executioners easily), and aside from Kled's massive base health, if Kled lands a Q on Vladimir, Vlad WILL be knocked back if Kled can maintain pace with Vlad. However, Vladimir is still much too safe, and despite him seeming weak and underwhelming in the early game, he's not just a cs machine, he's getting that cs for a reason, and that is his monstrous late game.
Either make his early game sustain weaker or hit his late game burst damage. One of those needs to go.