Do not overestimate Champion mobility. Night Hunter's bonus movement speed grants no bonus while running away from enemies, and Tumble doesn't go on cooldown until you actually attack an enemy which essentially adds 6 seconds of cooldown that you can't reduce with cooldown reduction when you are using it to run away.
Tumble can cross impassable terrain such as the small ledge near dragon pit if you position correctly.
Note that Tumble's extra damage only activates upon attacking enemy units, not structures.
You can escape in the midst of a battle by using Final Hour and Tumble to stealth and confuse the enemy. It is recommended to change direction than the one you were running in or to choose an alternate route.
Because she is squishy, it is highly recommended to opt for more defensive summoner spells, such as Ghost, Cleanse, or Barrier. Flash is very desirable. Final Hour in conjunction with Tumble, which only stealths for 1 second, doesn't make for a very good escape. These summoner spells can also assist in chases.
Building Sheen or Trinity Force can add some significant burst to Tumble. Starting with Phage provides much needed survivability and can assist in running from or toward a target if Vayne procs it on an enemy minion. Building Zeal first grants her added mobility and attack speed to more easily farm and proc Silver Bolts.
Vayne has zero abilities which scale with ability power, so consider this fact when deciding what order to buy items in.
A commonly held misconception is that due to Silver Bolts dealing true damage, Vayne does not need to purchase Last Whisper to deal with tanks, instead only needing more attack speed. This is somewhat not true, as most calculations with endgame builds (based on low ELO games) have True Damage only composing around 7-10% of her total damage. Moreover, an attack speed reliant build without a Last Whisper is somewhat easy to counter with items like Randuin's Omen and Frozen Heart as thogether, these items slow Vayne's attack speed by 23%, as well as providing a large amount of armor that Vayne does not have penetration for.
What is true, however, is that Vayne does not need to buy Last Whisper until late-game, unlike most ADCs, as enemies don't have significant amounts of armor early game, and sometimes even mid game. Silver Bolts will be enough to deal with them until late-game, where it's necessary to buy Last Whisper because most of Vayne's damage is still physical.
Some players opt to buy Youmuu's Ghostblade instead. This item provides an active that boosts Vayne's attack speed even more, allowing her to proc Silver Bolts a lot more times, not to mention the extra critical chance. However, the flat armor penetration may not be sufficient to deal with tankier opponents, so this item is ill advised if there is more than one champion stacking armor on the enemy's team.
Infinity Edge provides a substantial boost to Vayne's damage output from auto attacks. Remember that critical hits don't proc from Condemn and don't scale from Tumble's bonus damage.
Alternatively, getting an early Blade of the Ruined King can benefit Vayne due to the increased attack speed and incredible item active for damage and chasing/escaping. The passive effect also synergizes with Silver Bolts.
Enchanting your boots with the Furor enchantment allows Vayne to kite or escape very well. Tumble away and whenever the enemy comes within range of an autoattack hit them once to start the tumble cooldown and rush away with your furor bonus, tumbling once it's off cooldown. Furor also strongly boosts Vayne's chasing ability.
Choosing someone such as Miss Fortune or Caitlyn who have strong early games can easily deny Vayne from farming as Vayne has a relatively weak early game. Avoid pushing the lane as TumbleVayne makes her an exceptional farmer under turret, and has strong chasing potential.
Champion entire kit is focused on dueling (or 1v1ing) and chasing so deny her the potential to duel with you and her kit will naturally round out late-game when teamfights become dominant, where the fights become more stationary and targets become harder to focus.
Once Vayne obtains Final Hour at level 6, her dueling potential increases exponentially and she essentially "jumps" to her mid-game potentially immediately before most other Attack Damage Carries. The Attack Damage and movement speed boosts are huge given how early in the game it is and stealth detection is much harder to afford earlier on. Respect her presence at level 6 if you haven't pulled a significant advantage over her and avoid dueling her if possible.
Vayne is one of the most powerful duelists in game and is well known and feared as one of the most powerful marksman if fully build. It is critical to keep her early and mid-game crippled.
Marksmen are usually easy prey to one-burst-wonder assassins, especially Zed and LeBlanc . Vayne is by far not in this category, her ability to carefully move around the battlefield, drain-tank while evading attacks with stealth from Final Hour and bursting/peeling with Condemn makes her complicated to kill by surgical, tactical and careful assassins as she can often prevent their abilities to properly flow for a combo.
Silver Bolts is not that hard to counteract contrary to popular belief.
Champion kit provides extreme stealth detection and Cripple shuts down Champion mobility. Keep in mind that Cripple no longer reduces attack speed, although with the correct build, Lee Sin can usually one-shot Vayne late-game anyway.
Frozen Heart is extremely detrimental to Vayne because the armor portion of the item counters her lack of magic damage and the long-ranged -15% attack speed aura is practically unavoidable in teamfights. This item is arguably a better option in dealing with Vayne, as it does not provide HP, thus making Silver Bolts less effective.
Randuin's Omen is still especially useful against Vayne as one proc of slow will cripple her true damage and mobility. Because Vayne relies on her mobility in order to survive in teamfights due to her short range, the movement speed slow will make her an easy target to focus. As such, even building an early Warden's Mail will have noticeable effects against her.
The same case goes for lifesteal and high health regeneration; if the target's lifebar isn't noticeably large but is constantly renewed from either high lifesteal or abilities such as Sadism, the damage from Silver Bolts becomes significantly less noticeable.
Battling her only in teamfights forces her to switch targets, resulting in fewer Silver Bolts procs.
Mages with targeted burst damage/crowd controls such as LeBlanc or Ryze can easily dispatch Vayne as her mobility and Tumble are primarily anti-skillshot oriented. Targeted abilities that apply targeted hard crowd control's such as Infinite DuressWarwick and Nether GraspMalzahar prevents Vayne from retaliating with her mobility and lifesteal resulting in an easy kill.