The best way is to ensure you have the net margins to support the growth. Most companies don't charge enough to grow, so they keep taking on more customers - don't increase prices - and ask they're overworked employees to work more to keep up.
If you charge more, then you can hire more and deliver on a great customer experience.
Company culture comes from hiring the best talent and not filling a position with someone just cause it's easy. Be proactive vs. reactive. Define what you want, what the process for hiring someone looks like, and don't deviate. Involving the team in this process helps keep everyone accountable to the new hires, and ensure they support and cross train as needed.
Culture starts from the top. What you value, and how you make decisions will influence what others on your will do. If you want collaboration, then you need to collaborate with them. If you want everyone to take initiative and do great work, then you need to fire anyone who doesn't do this and tell them why - and fix your hiring process to not hire these types of folks going forward.
Growing pains are normal, so don't fear burning people out, as long as they know you have a plan and it's short term, most people will usually step up to the plate and help get over the hump.
Hope that helps.