Just like you do with any TV series or movie. It doesn't have to be exclusively Japanese (as some of them do include English words [Baka to Test to Shokanjuu] or completely use English words [e.g. Angel Beats! or Another])
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You wouldn't have to make up a name, you simply find a name you like and a viola!
Depending on where your character is from they may have a different name; for instance, if they are from Japan they would have a Japanese name (look at japanesetranslator.co.uk/your-name-in-japanese/ for translations of western names)
However if they are from other parts of the world try to find names that come from wherever they come from!
also the meanings of names might be important, espically if it's a main character and the plot includes them being heroic, or humble, or sweet and kind - so have a look at the name's origin and meaning too.
Try out a site like behindthename.com or name-meanings.com .
Dawn's last name is Hikari in the Japanese Pokemon anime, but in the English anime she doesn't have a last name..lol
If you are looking for an anime character that is a nurse, in Pokemon, the name of the (anime/manga) nurse is Nurse Melanie. :)
It came from the name akemi becuz that name is a animal
Anime
Try to go to different anime boards and asking people based on the description of the anime. Or you can also google specific names of the characters that popped up in the anime.