How does The Constitution give the president power to influence the legislative process?

I think so long as he influences Congress within the same boundaries as any other citizen, he's just excising his right to free speech. I mean, your question definitely raises legitimate debate. It is of dubious legality, I think. The president can probably influence Congress, again, just as much as the typical citizen can. As for forcing legislation down the throat of the Congress, I mean... any citizen could do that. It doesn't mean that their legislation would be accepted, but they absolutely can try. The president just has so much influence that he simply can. So again, it's of dubious legality, the debate centering around the presidents right to free speech -- where it ends.