I know that there is some legislation that President Trump has cited allows him to raise tariffs, as well as "fast track" that applies to multilateral trade agreements, but what allows him to exercise administrative authority in terms of bilateral trade negotiations (ex: China)? Would Congress have to approve or reject whatever he ends up negotiating with China?
Yes, Congress still has to approve, but "fast track" authority prevents Congress from sending back a completely different package that was "agreed" to which would, essentially, start the process again from scratch.
When there is fast track authority, Congress has to give it an expedited yes/no vote with limited discussion and no amendments, which gives any administration's negotiating team a lot more credibility when they agree to certain provisions in an agreement, since it will fly or fail by the language that is there, and they won't be revisiting the same territory over and over, unless it does fail to pass, as a whole.
Fast track is an expedited procedure for Congressional consideration of trade agreements. It requires Congress to vote on an agreement without reopening any of its provisions, while retaining the ultimate power of voting it up or down. The three essential features of any fast track authority are:
extensive consultations and coordination with Congress throughout the process;
a vote on implementing legislation within a fixed period of time; and
an up or down vote, with no amendments.