Wouldn't any changes to the tax code have to be first proposed and voted on by congress?
The first step is for the president to propose a budget. The budget will include revenue sources, i.e. taxes.
The second step is that the House of Representatives will create a bill making tax changes, often based on the presidential budget. If the House passes a bill, it goes to the Senate, which can make changes. If the Senate makes changes, it goes back to the House or to a special process called reconciliation.
Once both the House and Senate have agreed on legislation, it goes to the president. The president can veto, sign, or ignore the bill. If the president ignores the bill, it may pass without his signature or be pocket vetoed depending on the circumstances.
If the president actively vetoes the bill, it goes back to Congress to see if they want to override the veto (requires super-majorities of two-thirds of each house). If they uphold the veto, then the bill is dead and the process starts over.
Tax policy will be negotiated with Congress. A president does not have unilateral power to change taxes except in some edge cases (through regulatory interpretation). But the president's ability to veto budgets means that he has significant power in any confrontation with Congress.
I would consider "I will remove deductions and replace them with lower rates" as shorthand for "When I am negotiating with Congress, I will be pushing for a compromise that lowers rates in exchange for fewer deductions." It's true that a president can't unilaterally do these things. But do we really care if presidents make literally correct statements?
Note that both Hillary Clinton and Donald Trump use this same shorthand, as have past candidates (e.g. Barack Obama, Mitt Romney, and John McCain). The correct version sounds weaselly and doesn't make for good soundbites. The shorthand sounds stronger and is easy to quote.
As a practical matter, new presidents tend to have significant support in their first six months in office. While they can't keep every promise, they do tend to be able to accomplish most of two or three goals. For example, Obama passed a stimulus package (that he proposed), financial reform, and health care reform early in his presidency. The final results did not match his exact campaign promises, but in broad strokes, he did get those things passed.
Trump is unlikely to get the exact version of tax reform that he describes on the stump passed. If nothing else, it's short on details. But his version will tend to create the framework for whatever actually passes.