Here i have given one way to learn to recite this stotra , if you find any other way easier or comfortable please continue it and also let me know as it could be helpful for some one to follow and learn.
The way i learnt reciting Vishnu Sahasranama is by learning around 1-2 slokhas a day along with the meanings. In this way we can easily finish learning all the slokhas in around 2 to 3 months.
The following link has http://srikaryam.com/En/Vsns/Sthothram.aspx recitation as slokha wise.
Also for the full slokha recital at a time we could download MS Subbulakshmi’s recital (although beginners may find difficult to learn from this as Pronunciation is quiet fast).