As for history:
Texas won it's independence from Mexico the same way the USA did from England, armed up rising (rebelion) that ousted the heretofore dominate country.
Texas, as an independent country, but facing another war for independence with Mexico, petitioned the USA to become a State and was accepted. It was also granted the right to divide itself into 5 different states, hence a portion of Oklahoma, Kansas, New Mexico, Colorado, and Arkansas where originally part of Texas. In the case of Oklahoma every county west of and including Greer County was previously a part of Texas, the eastern portion was known as Indian Territory.
Arizona and New Mexico, or the most parts thereof, were purchased from Mexico the same way Louisiana was purchased from the French.
California, from San Fran south was originally part of Mexico, but after the Gold Rush the Americans slightly outnumbered the native Mexicanos. California voted to petition the USA to become a state, the petition was granted, and California (once a provence of Mexico) became part of the United States of America. It should be noted that a lot of Mexican citizens voted in favor of becoming a state of the USA, just as a lot of Mexican citizens fought against Mexico for Texas Independence.
For information the northern portion of California was once considered part of the Oregon Territory before Oregon (which included parts of Idaho and all of Washington) was officially designated a Territory, which occurred as a result of the California petitiion to become a state as boundaries had to be established amd the Shyskiu Mountians made a natural northern boundary.
What it all boils down to is that all of what was once part of Mexico, but is now part of the USA was accomplished legitimate means. Mexico has no legal claim to any part of the USA.