Internships can be a fantastic way to get your foot in the door at a company. Since you have gone though a selection process to get the internship, the company obviously already sees you as a good fit for the company culture. You now just have a trial run to see what you can do.
There are many things that you can do to get offered permanent employment after the internship. The overriding goal is to creating lasting value and this can be done in lots of ways. The problem you have however, is that you are short on time, so this can be tricky.
First things first - Who is who at the company
One of the first things that you should do in any new job, particularly one where you are short on time, is to create a map of who is who. The best way to do this is to take some time with your manager to discuss who works on what and for whom. To what level that you do this depends on the company. If you are working within a fairly big team you will only need to map out your team and people who your team report to. If the team you work in is fairly small then you should map out the people in your team and also who your team connects with.
Not only is this a great way to get to know people by name but you also begin to understand the company/department structure a bit better. Doing this with your manager or somebody you work closely with within the team will be really valuable to you but also show them that you are going out and understanding areas outside your day to day tasks too.
Understand what you do within your role
The best way to understand how you can add value is to understand your role. You will probably be given day to day tasks. This may include updating a system or process or producing some kind of work material. You should ask where this work goes, who uses it and what value it provides. This will give you an understanding of why you are doing what you are doing. Countless interns just complete their tasks without actually understanding why they are doing the work that they are doing.
By understanding why you do what you do you are then able to add value to the work that you create.
Talk to senior members of staff about the internship
Interns often get afraid when talking to more senior members of staff. More often than not, people in higher positions will be very willing to give their time to talk to new talent. This not only benefits them as they understand the motives of the next generations of the workforce but also the company, as you are more likely to want to come back as a graduate if senior members of staff have spoken to you about the company.
It is important to remember that most people aren't just there to perform a function, they want to enjoy their time in work. If you have to hand something in to somebody, unless that look busy, don't be afraid to stop by for a chat. If you're stumped for something to say - people love to talk about themselves. Ask them how they got to their position. You could even be cheeky and tell them you want to convert this internship into a job offer and you would love some mentoring/advice from them on how to do so. This not only gets you great insider advice from somebody that knows the company well but all also puts yourself on the radar of somebody high up within the company.
Look for areas that you can add value within the company
This is the holy grail of internships. If you have a skill set outside if your work remit, look for areas that you could add value to the company using these skills. Then ask if you can go ahead and make the changes you believe you can make. 99% of the time you will be encouraged to implement any changes, improvements or additions as long as it doesn't affect your ability to do your day to day tasks.
Areas you can add value could be anything from shooting a video for an event, helping make better use of social media, researching industry content to turn into some form of digital media or starting a blog for your department/company.
If you don't want a job offer, still try
Sometimes internships at companies aren't what you think they are going to be. So you sometimes don't want a job offer afterwards. However, even if you don't want a job, you will have a lot more to talk about when you go into other job interviews if you have changed a process or added value.