You're speaking about about pepole in general (basically all people) and not any specific people. When speaking about things in general, you use present tense, not past tense. This means you should say 'People become close...'

If you are speaking about specific people, you should put 'the' in front. So, 'The people became close' which makes sense. Then 'think' should be past tense.

'The people became close when they thought...'

Also, the first sentence should have a 'th' front of 'think'.

So, People become close when they think alike

and

People become close when they think the same way.

Finally, the first sentence is better because the words 'think' and 'alike' are often put together. 'To think alike'.

There is a saying - 'Great minds think alike'

By itself it sounds like an order or command so it should be said with please or thank you after it. 

Why I have started an account on/using italki.

This is my first time writing on italki. I´d like to learn english and improve my writing skills.

(The verb 'would like' makes the verb afterwards have a 'to' in front of it. 'I'd like to learn, I'd like to improve'. A normal 'I like learning' is okay. 'Improve' doesn't need a 'to' because the 'I'd like' is affecting 'improve' as well.)

I met a guy from London who talked/spoke (spoke sounds better) to me about this social media site. I always have problems passing writings exams.

(Writing doesn't need an s, because it is a participle, that describes nouns. Eg, running boys, talking mothers, slamming doors.) 

I need to practise more and more, to get enough levels

(grammar is okay for 'to get enough levels' but it doesn't make sense. 'To become/get good enough, to have a high enough level' is better)

to feel ('confidence' is the effect of getting good at English. 'So I can feel' is better than 'to feel' here) confidence with myself.

It would be a pleasure, if you could correct my mistakes in this simple text.

Thank you in advance

Oscar.

(Well done and Good luck. This is my first correction too, so I hope it isn't too confusing!)