It will be a tough sell
First you need a proposal. That means one of two things have to happen
1. Two-thirds of both houses of Congress vote to propose an amendment,
2. Two-thirds of the state legislatures ask Congress to call a national convention to propose amendments. This has never happened ever in our history
Then once all the arguing about the proposal is done, it has to be ratified. This means, again, one of two instances
1. Three-fourths of the state legislatures approve it
2. Ratifying conventions in three-fourths of the states approve it. This method has been used only once -- to ratify the 21st Amendment -- repealing Prohibition.
What are the odds of amending the Constitution? Pretty poor, and this is by design.
There have been thousands of proposals to amend the Constitution. Answers.com says 11,000 and that is probably a good enough number. In recent times, on the average about 200 each congressional term.
Of those thousands, only 33 have ever passed the "proposal" stage. If 11,000 is accurate, the odds have been 0.003%. After the proposal, the odds get a lot better
Of those 33, 27 have been ratified, but this also includes the first 10 (Bill of Rights), so the numbers are actually 17 (52%). Considering that one of them was used to repeal another, the odds drop down to 45%.
We would be much better off passing and
ENFORCING immigration laws. In fact, we don't need to pass any more laws or amend the constitution. Our best practice is to start enforcing the existing laws.
Only if after consistently enforcing the existing immigration laws, which includes punishing Americans who hire undocumented aliens, should we even consider a new law.