I would like to use Kanji for my first name (Winri) when I go to Japan. Is this okay to do? If so, what characters would go into the name?
Of course, you can use Kanji for your first name. I think your first name is ウィンリ written in Japanese so you can use kanji character like ウ, イン, リ.
There are many kanji and you can choose them as you like. I advise you to choose kanji which have good meaning.
How about 宇(ウ)院(イン)理(リ)?