Ok....and I take your using a Mac yes?
PS version?
Whatever.....going from 720x540 to 1920x1080 is a HUGE jump....whatever you do its not going to be pretty.
You need to make that image over 5 times bigger (266.67%!!!!).
The "marching ants" represent the size after up-scaling to fit the width.
You can try this in PS but the outcome very much depends on the content of the image.
1. Select the 'Image' layer in the layers panel.
2. Hit Command + T to enter transform mode...
3. At the top of the workspace enter 266.67% for both the width and height ( W: H: )
4. Accept the transform.
If the image dosen't look right try changing the "Interpolation" method.....you haven't said what PS version you use so I can't say where that is for you but normally when in "transform mode" (Command + T) its a dropdown option in the menu bar at the top of the workspace.
You may need to try several different Interpolation methods before one of them "looks" right.
Other than getting a larger version of the png there's not much else you can do.....there are only so many pixels in the image.
See how that goes anyway......you may get lucky.
Regards.
MrTom.