You are probably weak. Strength training is the solution.
-
The best option would be to learn to lift weights. Starting Strength, a 3-times-a-week barbell program, is a good option. StrongLifts 5x5 is also commonly recommended. (See [this question][1].) 

If you can't get access to a gym or barbell, you could look into bodyweight strength training instead. These [two][2] [questions][3] have some good resources on that front.

With More Information, You Get More Advice
-
Since you gave us your diet and dimension info, I now more firmly suggest strength training, and would add:

 - [BMI is not a valid metric for individuals][4]. Please don't misuse it, or allow others to misuse it, by applying it to people instead of populations.
 - Why do you want core exercises? I hope you have not fallen prey to the distressingly common virus called [Believing in Spot Reduction][5]! You can't get rid of belly fat with crunches or any other targeted exercise.
 - I am not convinced that you have a weak core, since you haven't stated any reasons for believing so. You might be weak generally, but as bodybuilders say, *everywhere* is a weak point if you don't lift.


  [1]: https://fitness.stackexchange.com/questions/2731/what-are-some-good-5x5-programs-i-can-follow
  [2]: https://fitness.stackexchange.com/questions/3714/bodyweight-exercise-resources
  [3]: https://fitness.stackexchange.com/questions/4578/what-are-the-top-intensive-bodyweight-exercises
  [4]: https://fitness.stackexchange.com/questions/193/how-many-calories-to-burn-daily-to-see-a-change-in-weight-within-a-month/3364#3364
  [5]: https://fitness.stackexchange.com/questions/1299/how-to-lose-fat-only-in-your-thigh/3933#3933