Create a folder in your project folder, and add your fonts to it. Example:

Add the below code in package.json.
{
...
"rnpm": {
"assets": [
"path/to/fontfolder"
]
},
...
}
For the example above, our package.json would now have a path of "mystuff/fonts":
"rnpm": {
"assets": [
"mystuff/fonts"
]
}
Run react-native link command.
Using custom fonts on project below code
<Text style={{ fontFamily: 'FONT-NAME' }}>
My Text
</Text>
Where FONT-NAME is the prefix platform specific.
FONT-NAME is the words before the extension in file. Example: Your font's file name is Roboto-Regular.ttf, so you would set fontFamily: Roboto-Regular.
FONT-NAME is "Full Name" found after right clicking, on the font file, then clicking on "Get Info". ( Source: https://stackoverflow.com/a/16788493/2529614 ), in the screenshot below, the file name is MM Proxima Nova Ultra bold.otf, however "Full Name" is "Proxima Nova Semibold", thus you would set fontFamily: Proxima Nova Semibold. Screenshot - 
Run react-native run-ios or react-native run-android again (this will recompile with the resources)