As a normal Yuumi support, you try to heal and buff your team. As Damage Yuumi Support you are one of the main damage dealers in your team. If you play right you will be able to do 700 + 8% current health every 5 seconds with your Q, an ability that is very hard to dodge if you are on a teammate.

But won't my team suffer from not having a proper support?
No. since your E scales really well with AP you will still be able to do a decent amount of healing even without support items. And the little healing your team will be missing out on will be more than made up with the amount of pressure you can apply with your Q NUKE.

You will be able to kill a squishy enemy with 2-3 Qs towards lategame and they can't hide behind minions or the frontline, this makes it so their squishy damage dealers will either have to stand way back in teamfights or one of their tanks will have to intentionally body block your Q every 5 seconds.

In the early game you will need to poke a lot, since you won't be able to apply any pressure with positioning or CC like Blitz or Thresh and you won't be able to outheal a dedicated healer your ADC is really vulnerable. To make up for this you will need to make sure that their support will never be able to comfortably engange by forcing their position in the lane with your Q. You will also have to poke down their ADC so that they are too busy with trying to safely CS to be able to poke or engange your ADC

Get Dark Seal ASAP and then work on getting as much AP as possible. You want to be able to outdamage their botlane as quickly as possible so your ADC can just sit there and farm.

The biggest threat to Yuumi and her ADC is the enemy jungler so make sure to ward plenty and ping your ADC if a gank is about to happen.

Since you mostly don't have to worry about your own positioning except for when proc'ing your passive you will want to keep an eye on the minimap and the overall state of the game. Your ADC has lots of things to worry about so try to help as much with the macro as possible.

Lastly, remember. You are playing DAMAGE YUUMI SUPPORT, no one expects you to do any damage. The enemy support or ADC will try to 1v1 you if given the chance, FIGHT THEM. As long as you can hit your Q skillshot without being on a teammate consistently you will win a lot of time, especially against their support.

Towards the end of the laning phase you have a choice to make, Should you stick on your ADC or should you hopp to another ally.

Usually you want to sit on the teammate that can get the most kills, preferably someone who roams. This ally is now your child, take care of them and let them believe that they are the reason they are getting tons of kills.

You will want to secure as many kills as possible without actually killstealing. Some people will get angry when Yuumi gets 10+ kills in a game but they aren't thinking about the fact that you are basically an unkillable second ADC, you need the kills as much as anyone else.

Do whatever you want
you have 1100+ AP

You can 1v1 most of the enemy team and win
You can kill a Squishy in 2-3 Qs without them even seeing where it came from