This is a lot easier than it may seem and I will try to explain why.
To start the Party the achievement refer to, you need to serve 5 customers
. Each have a portrait in the 5 middle lanes of the table and they are lit up once they are served. Followed by starting up the actual party mode which you can't "fail", it's just there for you to score big points.
This is all done within one mode of the game that you can start with 1-3 customers per time you start it. You start the mode by hitting the "Bob's Burgers sinkhole" 3 times which is located just to the left of the ramp in the middle of the table.
For every 4 times you hit the middle ramp you add 1 extra customer to the mode, up to a max of 3 (You always have one by default). So if you hit your shots you could do this in 2 modes. But no need to overdo it, you could just as well do them 1 by 1 if you want.
The actual mode have several stages of which all are timed
but for the achievement you only really need to focus on the last one where the portraits of x amount of customers are blinking and hit the ramp associated with them
for it to count them as served.
Everything else in between is mostly to increase the score you will earn per shot during the actual Party Time mode. So if you just want the achievement let the ball rest on a flipper until you reach the last part and hit the ball up correct ramp/ramps (from what I could tell it was random, but you can't get a ramp of a customer you've already served). If you miss and the time run out don't worry as you can just start the mode up and try again.
After you've served the 5 customers you only have to hit the Bob's Burger sinkhole once to start the Party Mode.
2 things worth mentioning
1. Increasing the order values during the timed modes increase the time you have to hit the ramps of the customers to make sure you serve them (thanks to frenchdawg for pointing this out).
2. The table have a day time mode and a night time mode, you can only serve customers during the day. If it turns to night you can speed back to daytime by holding down