At this point the "best" way is to use KIELER's KLighD and formats service. It's by far not the easiest nor a "minimal" setup though. So ...
There's no off-the-shelf solution at the moment. You may be able to copy a solution that Miro created for .kgt files. Have a look at this commit [1] and the API to convert json text into an ELK graph [2].
You'll have to copy and adjust these two files from the elk.graph.text.ui plugin (you can put the copies into the elk.graph.json plugin for testing):
org/eclipse/elk/graph/text/ui/layout/TextLayoutConnector.java
org/eclipse/elk/graph/text/ui/layout/TextLayoutSetup.java
What you have to do is likely the following:
1) Adapt the 'buildLayoutGraph' method in the 'TextLayoutConnector' to load the json text from the 'workbenchPart' (which is likely a 'TextEditor') and convert it into an ELK graph.
2) Adjust the 'TextLayoutSetup' and register it in the elk.graph.json's plugin.xml (along the lines of the plugin.xml changes in the commit)
3) Make sure you require elk.core.ui in the MANIFEST.MF
I'm sure I forgot something but let me know if give it a shot and have any success.
Regards,
Ulf