Learn English – “To do this” or “For doing this” to start explaining something

Your first example is idiomatic.

I did my best to do this. ... It means “I tried as hard as I could to accomplish this”—regardless of whether you actually did accomplish it.

Your second example is not idiomatic, but it is very close to a different idiom.

I did my best in doing this. ... It means “This was the best work I had ever done”—you achieved a new level of excellence.

I searched The Corpus of Contemporary American English (COCA) for both strings. Here's what I found:

  afford to go        182 results
  afford going        5 results

There's a strong preference for the infinitival complement, so I suggest that you use it in your own writing.

That said, I would accept the other version as grammatical. This is just my personal judgment, but Jim left a comment which agrees, so I'm not the only one. This may be part of a larger trend that some have called the Great Complement Shift, although at the moment the infinitive is still strongly favored; there isn't much evidence in favor of a shift to the gerund for this particular verb yet.

Of the five results in COCA for afford going, four are from the spoken language sub-corpus, and one is from the news sub-corpus. This, along with the relatively small number of results, supports the idea that it might be considered more colloquial.