EDIT : The reason why I advocate talking to the boss instead of the coworkers seemed to be unclear, so I'm editing my response to clear this up.
Personally, I wouldn't talk too much about this to my other coworkers. From what you describe, messiness is the norm in your office, people in general don't clean up after themselves. Even if some coworkers might prefer to have a cleaner work environment, you do work in an office where some people are okay leaving urine drops on the toilet seat for the next person to find. Those people won't be moved by a plea to help the cleaners out, and they probably won't like having their behavior monitored and commented on by a colleague, even in a non-accusatory way. Also, you risk being known as the one who "wants everything to be clean" instead of "the one who does kick ass analysis". If you're a woman, and are surrounded by men, sexism might even come into play. Since it is an office wide issue and there's a big chance coworkers will be annoyed by your remarks, it would be best if the incentive comes from above, your boss.
The only thing I'd might do is gently enquire around you, like in ElizB's response. If you find that some people are also bothered by the state of the office, you might go as a group to your boss. This will show your boss that this impacts a portion of the office, not just one person.
When talking to your boss, I'd focus on the state of the office, and not on the one dirtying it up :
I'd like to talk to you about our office space. I know that things getting a bit messy is normal, but from what I see our workspace can be quite unhygienic. [list examples]
Some battles you probably won't be able to (or shouldn't) fight. Some food crumbs on the balcony is normal. If you have already two baskets to recycle and people don't use them, you won't be able to force them. Even the overflowing trashcans, while disgusting, is tricky to fix1. Also, about the clogged sink, don't mention it if it has happened only one time. One time is an accident, it's not a pattern, so not a foreseeable problem. Focus on repeat offenders.
How you talk about it to your boss depends on what you know of him/her and your relationship with him/her. I'd be very candid, because I believe being kind and honest is always the best policy and is often the most effective, but I also know I can be very candid with my manager.
So, to my boss, I'd say something like this :
I'd like to talk to you about our office space. I know that things getting a bit messy is normal when you work together in one space for hours each day, but from what I see our workspace can be quite unhygienic. The sink has been clogged more than once, the toilets are often soiled with urine and the trashbins in the bathroom are often overflowing. Going to the bathroom can be quite disgusting. Also, there are dirty coffee mugs everywhere, which just gives an unhygienic feel to the whole workspace. Again, I know some of this is normal, and kitchen and bathroom etiquette are difficult things to manage in every office, but could we do something to reduce the volume and frequency of these incidents ?
With this script, I hope to convey my complaints while seeming reasonable ("I know things get dirty in offices, but we seem to get more dirty than the norm"). Also, I want to know at the end of the conversation if my boss agrees with me and plans on doing something. Because if he doesn't, you know this won't change. You have your answer.
Also, know that just because your boss agrees with you doesn't mean things will magically get better. Bathroom and kitchen etiquette are the most troublesomes to fix. The advice column Ask A Manager has a lot of posts on this subject (here's one). You'll also see in these kind of posts people's annoyance with colleagues who police their coworkers' behavior (again, why I'm for talking to the boss instead of talking to colleagues).
1Aside from asking the cleaning company to come more frequently, or assign a coworker to do this