To show you're ready for bigger things, take initiative when working on your current tasks. If you haven't been told what to do next, don't just take it easy or wait for instructions. Actively think about what the next steps might be and take action (but do get approval if appropriate to the situation). Go the extra mile to make your work not just acceptable, but outstanding.
In team discussions, take the lead to keep the group on the right track. Be careful not to interrupt anyone or come across as overbearing. Being a leader in the discussion doesn't mean talking all the time. Ask the right questions and make useful suggestions so the meeting is productive. Encourage ideas and input to generate team enthusiasm for the project.
Ask for feedback from your boss and from colleagues to show your willingness to improve. Find out what you can do to work better with those on your team and how you can contribute even more than you are now. Be open to listening to ideas on how to grow, even from those who are less senior to you.
You should also be open with your own opinions and suggestions, giving feedback to others. Frame it in a positive way, not as a criticism but as an opportunity for growth. When you see a solution to make things run more efficiently or you have a new idea for a product or campaign, don't be shy about bringing it up. If you're consistently making positive contributions to the company, you'll begin to stand out as a leader and push yourself up to the top of the list when promotions are being considered.
Take the lead in team discussions
Show your willingness to improve
Voice your opinion