For undergrads pursuing a career in consulting, there are two primary 'feeder' tracks most large firms recruit for:
1. Technology Consulting
If you have a computer science or similar quantitative background along with the ability to solve problems and communicate with impact, you would be a strong candidate for an entry level position in an IT consulting role. Because consulting is a client-facing professional service business, your communication skills along with your emotional intelligence (EQ) will be just as important (if not moreso) than your technical ability. Consultants are also expected to demonstrate a high degree of professional maturity as in many cases you will be interacting with clients who are significantly older and more experienced with a corporate setting. This will get tested in your campus interview through a series of behavioral interviews which test how you respond to situations such as what you would do if a client asks you a question you don't know the answer to.
2. Management / Strategy Consulting
This track tends to be more difficult to get into as an undergrad because unlike those when strong technology backgrounds, you need to demonstrate the ability to solve unstructured problems. It is more difficult to demonstrate this in an interview vs. demonstrating you understand technology. In my opinion, it would be more beneficial that you understand how companies operate, what challenges they face, and how those challenges are typically solved. In other words, a strategy/management consultant needs to be able to solve problems that might require a combination of business process change, organization change, technology change, or a combination of the three.
So what experiences do you need?
First, any experience you can get that allows you to talk about a situation where you solved a business problem for a real company would be great. Some practical ways to do this are through your school's consulting club as was previously mentioned. Another option is to perhaps find a local business and approach the owner about volunteering to help them with their business in an area you are passionate about.
For example, let's say there's a local apparel store and they want to start selling online, but a struggling to keep up with what inventory they actually have to sell since it's going out of the front door with in-store purchases and out of the back of the store through shipments from online orders coming from eBay store, etc. Perhaps your 'project' could be to research solution options based on their needs and develop a presentation where you walk through through a thoughtful analysis of how you approached your research, what options your reviewed, how they ranked vs. your 'client's' needs and then a final recommendation.
If you have a technology background, another example would be to take the same approach and see if you could a side project leveraging your technical skills. This could be anything, but if you could demonstrate awareness of typical phases of a project (Design, Build, Test, Deploy) that would similarly help you articulate this in an interview setting.
While these are small examples, they demonstrates passion, 'hustle', and give great real-world experiences that might help you better understand what a Fortune 100 company might go through.