Nihilism (/ˈnaɪ(h)ɪlɪzəm, ˈniː-/; from Latin nihil, meaning 'nothing') is the philosophical viewpoint that suggests the denial or lack of belief towards the reputedly meaningful aspects of life.
Let's try to see this problem from different perspective. Let's try to express emotion first. Imagine if your character is mad, for example. What would you do to express that to the readers? Of course you wouldn't want to just state explicitly that she's mad, as you seem to have understood. Instead, you would want to explain why she's mad. Maybe she was cut in line this morning, maybe her boyfriend cheated on her, maybe she found out her gucci bag was fake, etc.
But you need more than just reason. Imagine without context you said in your novel, "She found out her gucci bag was fake". You might expect readers to get the impression that she was upset. But upset is when you have expectation on something, you want it to be true, but then it betrays you. Who knows if the readers thought she accidentally bought the super expensive bag, so she actually hoped for it to be really really cheap high-tier fake bag?
The solution is that you need to also tell readers all of the variables that entails the referred emotion, which is whether she expected it and whether she wanted it. If you want the character to be upset, you need to tell readers that she thought the bag was fake, but she wanted it to be authentic. Now that the readers understand that, when it's revealed that the the bag is indeed fake, without explicitly saying that she's upset, you can let the readers make sense of it on their own. Of course people will still have different emotional response for the same situation, so you still need to define first what makes her feel certain emotion.
Now you can try to apply this formula to express your character's nihilistic point of view. First we need to define what nihilistic is, and recognizes all of its variables. I would say that you are nihilistic when you see something as having a really noble purpose, but then you find other thing that appears to completely nullifies that. The variables are what what the noble purpose is, and what nullifies it.
If you want to make sure the readers understand that she also feels depressed, empty, or angry, or scared, or confused, you just need define what those feelings are, and add the corresponding variables into the explanation. For example she recognized that the waiter has been extremely polite to customers this all this time, but then she realized that people never cared about that attitude, so there's no point of being polite at all. Now if you want her to feel angry about it for example, define angry first. Maybe angry is when you feel someone or something has deliberately done you wrong. You then need to assign value to those empty variables. Maybe you want to feel that everyone has done you wrong by not sharing the same value which is appreciating politeness. Maybe you want to feel that the waiter has done you wrong by not wanting to change his attitude.
Some of other examples would be like "The waiter is really cute, but this feeling is just atoms interacting in predictable manner" or "The waiter is really cute, but there's a high chance that he already has girlfriend" or "Those people are always trying to pursue career, but no one wants to have kids and the country's birthrate has been declining" or "Everyone just wants to be happy, but there's a small chance of earth getting blasted by gamma ray burst from a nearby quasar" or it can be as simple as "This coffee is very delicious, but it's just 98,75% water, the remaining portion consists of non-essential substance for my body, and hanging out is definitely more fun than this."
If you want the nihilism to be existential, you just need to change the definition a your nihilism a little bit to be "nihilistic is when you see something as having a really noble purpose, but then you realized everyone is going to die, which completely nullifies that."
Also, you don't need to be very methodological like this in approaching your work, but it's good to understand how things work a little better.