Hamlet didn't bring the poison to the fencing duel, or put it in the cups. However, Hamlet killing Polonius scares Claudius; he wonders if he might be next. This inspires him to engage in the Rosencrantz and Guildenstern scheme, which fails. Then, Claudius goes the whole nine yards with the poison scenario, in order to ensure Hamlet's demise. At the ending, Hamlet just took advantage of the cards he was dealt. There's poison here? Well, I guess I can use that to my advantage and finally enact the revenge I have been ho-humming about this entire time. So, he does. Hamlet's mother was unfortunate collateral damage in the entire wreck that was the battle between Claudius and Hamlet. So, there is a connection to the death of Polonius, but is that the definite beginning? You could say that the murder of the king, or the ghost appearing, or the play could have set all of this going.
Also, who's to say that if Hamlet hadn't killed Polonius Claudius wouldn't have plotted Hamlet's demise? Wouldn't the suspicion of Hamlet knowing about the murder have been enough to prompt Claudius to scheming? Claudius had a lot to lose if his secret was outed; the kingdom, Gertrude, honor, and his life. Polonius or no, there is a good chance that events might have ended badly anyway.