Yes, you can, but you'd better have deep pockets! First off, it's no surprise that your firm is not ranking highly for the phrase “New York personal injury lawyer,” for the simple fact that this is one of the most competitive search strings in the American legal system—and there are only 10 slots available on the first page of Google results. Even if you do absolutely everything right, it might take years for your law firm's website to claw its way near the top—and even then, you could be taken right down again by the next Google algorithm change.
Now on to your question about a pay-per-click campaign. The way this kind of advertising works, you will pay more when prospective clients click on more general key phrases—and “New York personal injury lawyer” is about as general as it gets. Don't be surprised if chasing this phrase costs you hundreds of thousands of dollars, which might be better spent on a different kind of web advertising or a redirection of your marketing efforts toward TV, radio or print media.
However, if you're feeling less ambitious, you can spend a fraction of that amount on more highly targeted phrases with lesser search volume. If “New York personal injury attorney” is out of the question, you might want to consider something like “Brooklyn car accident lawyer”—or even get more specific (“Flatbush fender-bender attorney” probably won't set you back all that much). Better still, you can purchase a wide assortment of less popular key phrases for a fraction of the cost of “New York personal injury lawyer.”