To do one’s utmost is to make the maximum possible effort: We will do our utmost to meet the deadline imposed by the regulator.
What one sometimes sees (or, more often, hears) is upmost instead of utmost. Close, but no cigar.
Upmost is a legitimate word, but it’s obsolescent in its correct sense. It refers to something that occupies the highest place or most important position.
The more common word for that is now uppermost. As in, The reader’s convenience was not uppermost in the mind of the drafter of these complicated regulations. Or, The book was on the uppermost shelf in the library.
An example of the correct use of upmost is found in Decision No. 2399/14, 2015 ONWSIAT 3, which refers to the ‘upmost flexion’ of the distal phalangeal joint (at para 8). See also Re Loblaw Groceterias Co. Ltd. and Minister of Highways, 1963 CanLII 222 (ON CA),  1 OR 271 (the ‘upmost round’ of a ladder).
The remaining 92 cases in CanLII which use upmost get it wrong (upmost good faith etc.): utmost would be correct. In only a handful of those 92 is there a judicial ‘[sic]’ to indicate that the error occurs in quoted material.
Avoid upmost entirely – especially when utmost would be correct. Use uppermost when you mean something like ‘chief in importance’ rather than ‘best efforts’. For those, utmost is the way to go.