Bel'Veth is the newest champion added to league with an astounding presence and popularity with her.
She is a late game fighter with a weaker early game, but still hard to kill if you manage to do it right that can easily pressure lanes after killing rift or baron in the process.
She's cemented herself in S+ Tier as the 8th highest winrate jungle, soon to be 7th passing Wukong at the rate she's going in every rank.
It'll be safe to assume in patch 12.12 she's going to be nerfed. How? More than likely her lifesteal above all things and her attack speed may take a hit too. Enjoy her OP power while you can, It's been confirmed she's gonna get a nerf but no numbers have been given yet on how.

Bel'Veth, while new, has proven to be a very useful champion in the longrun as a lifestealing fighter with scary potential to hard carry if given the chance. A hard learning curve but a decent to understand kit, Bel'Veth is a champion anyone can pick up and learn but not easily master. I haven't obviously had a ton of time with her, but I can tell you with certainty she's going to be a good champion in most situations.

Bel'Veth is definitely a good option as a damage dealing, life stealing empress who has very good scaling power in terms of damage and is a scary champion if given a chance to be placed ahead.
1v1s are her specialty mid-late game and makes her scary to fight without grievous due to her massive lifesteal and unlimited attack speed cap. Seeing you means enemies can get scared to fight you if they aren't smart enough to avoid you or if your laner is good enough to help you.

-Scaling Machine: Bel'Veth has some heavily good scaling power that doesn't fall off too easily unless you let it. With her no-limit attack speed, she's a force that can't be easily overpowered in terms of 1v1 fights if done correctly, and her lifesteal doesn't let you take her down any easier.

-Good For Gank Pressure: Level 6+ is insanely good for lane pressure as her ult form increases her power overall and can especially be useful if you take rift/baron as minions who die in her presence when she takes their corals will turn into voildings that push the lane. Not to mention, her Q gap closes and her W knocks up leaving enemies in a vulnerable state to your teams laners.

-Fast Waveclear:Her Q provides her with insane clearing against wave minions and jungle monsters alike. She can easily go through most camps before scuttle comes up with her damage and clearing capability so she can easily get CS and get to ganking, then right back to CSing to the next lane pretty fast.

-Very Mobile: Her movement speed is impressive as it is, but not just that, her Q allows her to be pretty mobile being able to dash twice in one direction, and over walls in her ult form. It's good for certain ganks and lets her overall just be nuts for chasing and dodging.

-Weaker Early Game: Her early game is not the best, while she's good at clearing, her fighting power isn't the best early without help or being a bit fed. Once she hits 6+, her fighting power is kicked in, but around then even with her ult form active it's still risky.

-Weaker when behind: While it's true for most champs, this is no different for Veth, in fact even more true above anything. If she is behind, she is insanely weak and has a hard time catching back up. She's not a very useful champ overall if she can't keep up with her team or the enemy.

-Bad against CC Comps: If the enemy team has heavy CC, she's not very useful. Veth can dodge all she wants, but if she cant move, she cant interact or fight against an enemy team well and will just get melted in the end. She needs to move and auto to be useful and if both auto range and movement are taken from her, she can't be effective.

-Ult Form Reliant: While it's true, she isn't an easy 1v1 opponent in most cases, it's pretty much due to her ult form for this reason. She needs her ult form active when fighting in order to get the most out of her power and kit. If she is caught out without it, you won't exactly die, but you'll be in a weaker state than normal.

Bel'Veth has a safe early game but hard 1v1 late game playstyle. Her ganking potential is good but only if you can get a laner who's willing to help as not all laners will assist you in a fight if they aren't confident in themselves or you.

- Early Game: Your job early game is to farm up and give ganks to lanes who you can assist if they're willing to help you in the fight. 1v1s are possible but shouldn't be a priority until you hit at least level 6. Once you do that and get a dragon, rift or champion kill and pick up a coral, your 1v1 power just got alot better. Try your best to get fed and keep up, while helping your team do the same as well

-Mid Game: The Mid-Game can be insanely iffy, however as the jungler, obviously you're meant to get objectives and roam to stop any advances or help any lanes still hanging around in their laning phase. You have pressure if you kill rift and get its coral, so try your best to get it if you wanna apply lane pressure with your voidlings, laner and your own power.

-Late Game: Always situational. Your power is at its peak in the late game, so never be afraid to fight unless you're against a counter or overpowered by the enemy teams advances against you all together. It is your job to finish getting objectives, and picking off lone champs who try to play by themselves. Don't underestimate them or overestimate yourself if you know they're a carry or are good at their job. Especially watch out for baits. You're a big player in the late game so an early death means alot of pressure went out the window.

Pathing for Veth is pretty straight foward and consistent honestly without much need for change unless level 1 invasions are going on against you.

You'll normally want to start off at red buff on either side and take it. It'll be a quick clear as long as you have help. Once you get your red buff, go to krugs and smite the biggest one when it's closer to death, and use your Q to clear them out faster.

Afterwards, go to raptors [you can even go there second if you prefer to skip krugs] and use your Q to clear it. All 4 Qs landed on the entire wave will kill the smaller raptors for an insanely fast clear. Finish them, and kill wolves. They're pretty fast to kill thanks to your Q as well.

Next part is optional preference: You can either go camp the scuttle when it's about to spawn, or you can go get your blue buff first. If you go scuttle first, you can either W it, auto it then smite or you can smite it and use your Qs and auto instead to save your W for if the enemy jg is coming towards you.

After scuttle, kill blue buff/gromp depending on what you did, and check midlane and toplane if on blue side or midlane and botlane if on red side to see if either are gankable. If one is, you can try to go for it as long as your team isn't too low and is willing to help your gank.

If no lanes are gankable, go back to farming up and seeing if the other scuttle is taken or not to give you an idea of the enemy junglers position on what side of the map they're at. Once you do that, you can farm any camps that spawn, or back and get your items. Afterwards, it's all preference from there on your pathing.

Bel'Veths passive, Death In Lavender has two parts: Firstly, if she kills a large or epic monster, or kills a champion, she gains a stack of permanent attack speed the entire game. The second part allows her to temporarily gain bonus AS after using an ability which is seen in Lavender stacks under her health, telling her how many autos have bonus attack speed left. This is what makes ganking and taking large camps/objectives so important on her. She needs it to keep her attack speed growing as she has no attack speed limit, but does not gain attack speed when leveling.
Bel'Veths Q, Void Surge is her most used and main ability. Bel'Veth grants herself 4 dashes, 1 in each direction, dealing 10-30 damage off ability level +1.1% of attack damage. Each directions CD scales off her attack speed. This is her mainly used ability for damage, chasing, escaping and activating her passive. It is also her main wave clear tool in lanes and jungle camps, do take note though:
-Bel'Veth CANNOT dash over walls in her normal state. She must be in her true form to Q over walls.
Bel'Veths W, Above And Below has Bel'Veth slam down her tail, dealing 70-230 damage based off level, and 125% AP scaling and 1% per bonus attack damage. The slam knocks up enemies for 0.75 seconds and slows them for 1.25-2.25 seconds as well. This ability is not just good for gap closing and CC, it's good for giving you another dash. If a dash is on CD and you hit an enemy with a W, your dash in that direction will refresh and come off CD.
Bel'Veths E, Royal Maelstrom slashes in an AOE around her targetting the lowest health enemy, granting her 70% damage reduction, and giving her major lifesteal scaling 100% off lifesteal items. The damage also scales off the opponents missing health increasing the lower their health is and duration of the ability scales off attack speed.
I have a few point for this ability that should be well known:
-Firstly, every slash will proc conqueror, meaning your conqueror will hit its 12 stacks + healing mark near instantly as long as the enemy is in the zone
-Second, it can be interrutped, and I only recommend using it at low health if the enemy has used CC or you need it to execute
-Finally, THE MOST IMPORTANT THING ABOUT THIS ABILITY, you can use it on enemies who are hiding in brush or are invisible. So a Qiyana in her grass Q zone, an ulted talon, any type of invisiblity or hiding abilities will not stop enemies from being targetted by her E meaning you can still damage them even if you can't see them.
Bel'Veths Ultimate, Endless Banquet has a passive and active to it.
Passively:
-Every second attack against the same person will deal additional 4.5-7.5 [+0.9 bonus attack damage] true damage to enemies that stacks infinitly. Enemies you kill or get an assist against, or any epic monster kills, leaves behind Void Coral that can be consumed and grants her true form, giving her 100-200 [+90% AP and +1.65% bonus AD] max health, 25-75 movement speed out of combat, 50-100 attack range, 10-20% attack speed.

Actively:
-The void corals left can be consumed and grant 1 lavender stack, granting you her true form for 60 seconds. If it is a rift monster, the duration is raised to 3 minutes and spawns voidlings if minions in lane die in her presence. Anyone in her zone while consuming a coral are slowed, then she explodes, dealing 150-250 [+100% AP and target missing health damage] true damage. She also heals for 100-200 health.

Her ult is what makes her such a fearsome 1v1 opponent. The increased stats make her insanely dangerous to fight and is KEY to making her 1v1 potential at its peak.

Conqueror is probably one of the best runes tied for lethal tempo on Veth. Not only does it increase her damage, she gains a ton of healing off of it was well when fully stacked, which is easy to stack with her E, meaning she gives off tons of damage, heals off it and provides her with that extra power combined with her ult form to give her insane potential for fights.

Triump is good for giving you an extra bit of healing after killing/assisting an enemy champions death and provies you with 20 extra gold along with the kill/assist gold for giving you an easier way to stay in the fight longer. Overall it's just helpful for sustain and gold gain.

Alacrity is insanely useful for stacking up onto your already no-cap limit attack speed, providing you a faster gain in AS and overall just making you build it faster for fights overall.

Last Stand works very well on Veth as you'll be fighting for your life regardless, and if you somehow manage to gain lower health, your E will gain a ton of bonus power not only with lifestealing your health back to full, but also doing more damage since you're low, and doing a ton of damage to them based off missing health. It's a good combination to put up a fight if you're too low against someone.

Sudden Impact is a preference, but one I think is good for engaging. It grants you 7 lethality/6 Magic Pen for 5 seconds which can be useful for that extra engage damage. While it does have a 4 second CD after usage, it is still good for engages to boost your power up in those scenarios thanks to your Q

Treasure hunter is a good rewards system for ganking and killing champions, providing you with extra gold stacks every time. It's good to start getting you a bit more item fed early and letting you overall just do better at getting what you need to start scaling.

ALTERNATIVE CHOICES
Lethal Tempo is tied for Conqueror as Veths most useful main rune. While Conqueror gives extra healing and damage, Lethal outputs that damage at a high rate. Since you have no attack speed cap, late game this will be fully stacked insanely fast and continue to output damage at a rate that seems too OP

Eyeball Collection is a good substitute for Treasure Hunter if you'd prefer more AD than getting extra gold on kills. Giving you that boost in damage for killing enemies and making sure you don't have an easy fall off.

Good for giving yourself Out Of Combat movement speed so that you're capable of roaming and getting to objectives much faster overall. You already gain this kind of movement speed in your ult form, but this will just make your roam even faster

Magical footwear gives you free boots at 12 minutes reduced by 45 seconds per kill you gain on a champion and provides +10 movement speed than regular or tier 2 boots. This will be a useful way to get free boots, save up gold a bit better and provide better movement speed to keep up with enemies or roam faster

Cosmic Insight grants 15 summoner spell haste and 10 item haste. This is most useful for things to be on a lower CD such as BOTRK or Shieldbow if you need it on a lower CD for fights, and gives your abilites a lower CD for faster use in fights.

Bel'Veth is a heavy AD scaling, very useful member of most team comps that can give good pressure if used right and is an insane 1v1 god in the later game against most champions you may come up against.
Hopefully this guide gives you a better insight on exactly how to use her, her placement in the game and maybe a better view on just how Veth can impact the entire game with her power.