In Ms. Pelosi's sentence,
Mr. President—Mr. President, please don’t characterize the strength that I bring to this meeting as the leader of the House Democrats, who just won a big victory.
the phrase "as the leader of the House Democrats" attaches to "I" (that is, to Pelosi), and the phrase "who just won a big victory" attaches to "House Democrats" (that is, to the subsection of the Democratic Party elected to the U.S. House of Representatives, where the Democrats achieved a net gain of 40 seats over the number of seats they held in the previous session of Congress).
The part of Pelosi's sentence that is most difficult to make sense of is her choice of the verb characterize. The point of her response to Mr. Trump is to urge him not to imply that her quest to become Speaker of the House (by vote of her fellow House Democrats) in the coming new year severely limits her freedom to bargain with Trump over building a wall along the U.S.'s border with Mexico.
The implication of Trump's comment is that Pelosi's political vulnerability to intraparty opposition if she is seen as being too accommodating to him on the wall issue makes her weak as a potential bargaining partner in negotiations with Trump: she can't deal with his proposals on their merits because she is constrained by her precarious position as a candidate for Speaker of the House.
Pelosi's response is that the Democrats just won a significant House majority after having been the minority party there for the previous eight years and that Trump should refrain from speculating about her supposed political vulnerability at the (televised) meeting he was having with her, Vice President Mike Pence, and Senator Charles Schumer.
It seems to me that characterize is a rather odd verb for Pelosi to have chosen. She seems to be using the verb as a short form of a more complicated idea along the lines of "attempt to characterize in a denigrating way":
Please don't attempt to characterize in a denigrating way the strength that I bring to this meeting [following the House Democrats' electoral victory].
But it would have made even more sense for Pelosi to have used the verb mischaracterize (that is, "inaccurately describe or represent") in place of characterize:
Please don't mischaracterize the strength that I bring to this meeting [following the House Democrats' electoral victory].
Of course, Pelosi was speaking extemporaneously, in response to a provocative remark from Trump. If she had had time to edit her remarks for publication, I suspect that she would have replaced characterize with mischaracterize.