Short answers, pulled from the story.
Thomas Jefferson used the phrase Empire of Liberty in 1780 while the American Revolution was still being fought. He envisioned this empire extending westward across the American continent as crucial for the nation's future.
The United States acquired the Louisiana Territory from France in 1803 which almost doubled the area of the Republic and removed barriers to western expansion. Jefferson wrote to Dr. Joseph Priestley on the 29th of January 1804 stating he looked to this duplication of area for the extending of a government so free and economical as ours.
The Monroe Doctrine emerged as a U.S. foreign policy initiative introduced in 1823 that stated European efforts to colonize or interfere with states in the Americas would be viewed as acts of aggression requiring U.S. intervention. The doctrine was invoked during the Second French intervention in Mexico and with the German Empire during the Zimmermann Telegram affair in 1917.
American Protestant and Catholic religious activists began missionary work in pagan areas from the 1820s and expanded operations worldwide in the late 19th century. Among Catholics the three Maryknoll organizations were especially active in China Africa and Latin America while Jean-Paul Wiest wrote about Maryknoll in China covering 1918 to 1955.
Economic dimensions of the Empire of Liberty involved dissemination of American management methods such as Taylorization Fordism and the assembly line alongside technology and popular culture including film serving as tools for global influence. Richard Pells explored From Modernism to the Movies showing the globalization of American Culture in the Twentieth Century published by Cambridge University Press in 2006.
Jefferson initiated Indian removal policy through energetic efforts to obtain native proprietors' land along the left bank of the Mississippi River which resulted in aboriginal inhabitants' lands being drastically reduced through treaties shot through with fraud bribery and intimidation. He grimly predicted to John Adams they would relapse into barbarism and misery losing numbers by war and want when Indians interfered with white definitions of national interest as backward tribes did in 1812.