"My company does not have a formal process for giving employee reviews. What is the best way to get manager feedback and to discuss raises/compensation?"
Don't you know?
Annual reviews have fallen out of favor recently, with both Fast Company and Harvard Business Reviews covering the decline of this method of performance evaluation. It makes sense. How can a review that happens once a year accurately reflect the quality and scope of all of your work? It might be hard to get your entire company to adopt an annual performance evaluation, but that doesn't mean you shouldn't have some way of tracking your progress with your manager.
Talk to your manager
Tell your manager you want to have a plan that outlines what skills and achievements you need to grow into the next role. Set goals together and track the results. If you hit these mutually agreed upon goals, you will then have the evidence you need to request promotion and pay increase.
Meet regularly A healthy employee-manager relationship is one that includes regular feedback. Do you have weekly or bi-weekly check-ins with your manager? You should. Not only for the development of your personal relationship with them but also to track the progress of your goals, "direct reports of managers who check in on progress towards weekly goals are up to 24 times more likely to achieve them." - Fast Company.
Think about personal development
Outside of being able to track how well you're doing your job, you should also work with your boss to grow your skill set. You don't want to work in a "business-as-usual" environment. You should feel challenged and excited about the work you do, and usually this occurs when you're stretching beyond your abilities.
Maybe it's time to ask When was the last time you received a raise, promotion or bonus? If it's been more than a year, then you should feel comfortable approaching your manager to ask for a raise. A standard cost of living increase is about 2% per year. If you've taken on more responsibilities or if the scope of your job has grown since your last pay increase (if any) then you should also feel comfortable asking for a pay increase.
Your manager should be excited to see you showing initiative to grow within the company. If regular check-ins or reviews aren't met with glowing enthusiasm, don't worry too much. You're asking your manager to take on an active role in your career development (which they should be doing) and if it's unnatural to them, then it might take some time for them to feel comfortable in their new role.
If you try all of this and it goes nowhere, then it may be time to find a new job. You don't want to stagnate, and being in an environment where you are actively learning and growing is good not only for your personal development, but also for your salary.