The tenses are fine: You are writing now because you thought something in the past. It would be grammatical to say I think the best way... but this is not very idiomatic, and sounds rather pushy.
There are other problems with your prose, though. The first sentence is missing a subject. At a minimum, you need to insert a comma and I before thought, but it would be better to split up the sentence, perhaps "I would like to be a part of your team; I thought the best way was to ask directly" (assuming the first lacuna is the name or description of a job).
And the prepositions in your second paragraph depend on what exactly fills the blanks. Your background may be with IBM, in electronics or possibly at Harvard; too general a question misses out the context needed to give a good answer.