How to answer those questions in an interview so firm would hire me?

1. Tell me about yourself
2. What are your career plans?

RMS and an asset management firm asked both

On the first one, just answer honestly; it's not a trick question, the interviewers are generally just looking to get to know more about you that's all -- remember that the person interviewing you is probably going to be either your boss or someone you'd work with quite closely if they offered you the job, so the question they'll have in mind is "Would I feel comfortable working with this person every day?", "Will we be able to have a good working/professional relationship?".
Keep it positive and use this as an opportunity to show your best side, questions like this should be enjoyable to answer in an interview because you can reflect on the best things about 'being you' and try to build up a bit of a rapport with the interviewer (Because you will need to work with them as well - it goes both ways).

If you're struggling for ideas consider questions such as:
- What do you feel are your best personality traits?
- What motivates you and what you enjoy in life or are passionate about?
- How did you arrive at some of your biggest life decisions so far (e.g. why you made your A-Level or University choices)?
- What are you most proud of? (Maybe that's something you're involved in doing now, or something you've done/achieved achieved in the past)

On the second one about career plans, you don't need to have your entire future mapped out, but they will generally be looking for you to have actually spent time thinking seriously about your future, where it might lead, and most importantly how the job that you're applying to fits into it. Try to imagine where you'd expect to be after a few years in the job - e.g. what skills would you'll have hoped to learn, what kind of roles and responsibilities you'd hope to be moving into.

The most important thing on the second question is to be prepared for it. If you answer this question with words to the effect of "I don't know", then the impression that the interviewer is going to take away will likely be "I don't care enough to have thought about it", which isn't a good message to send. Employers like people who have some drive and ambition to build a career, and who is going to make the effort to learn and progress if they get the job.