"I am thinking to invest" or "I am thinking investing"?

I wouldn't use either one; I'd say "I'm considering investing in stocks". One does not "invest into" anything (in my experience). The first sounds at best somewhat stilted.

Your first sentence is a fairly common structure, but it is somewhat informal, to say the least. I would not recommend non-native speakers experimenting with such non-standard forms.

The second sentence is simply ungrammatical.

The standard phrasing is...

I am thinking of investing in stocks.

When you are considering a possibility or advantages of doing something, you use think of.

He was thinking of becoming a zoologist.
I am thinking of investing my money.
I am thinking of investing in stocks.

"I am thinking investing into stocks" needs a second part to make it grammatically valid. For example:

• What's up, John? You seem absorbed.
• I am thinking investing into stocks is a risky thing nowadays. I'll cancel the deal.

Which is contracted from "thinking that/[of] how investing...".

But, in general, there are better choices to express the same meaning.