If you wish to understand what "Righteousness" is - righteousness always leads to salvation. Righteousness is not concerned with proving itself to be good, awesome, or cool - Righteousness does that which is right and proper - and it is right and proper to save, to show love, even at cost to yourself.
Consider: Let us say you see a child playing in the street, and there is an on coming car. What do you do? Do you say, "My mother taught me better than the play in the street - getting pasted will serve the kid right." Do you say, "Oh, this will be sad, but really, it doesn't concern me?" Do you say, "Let the car come and decrease the surplus population?" Or do you try to get the kid out of the road, even if it means you get hit yourself?
It is just right to save, to rescue the kid, even at cost to yourself.
Now, this is I think easier when it's just a simple, quick reaction from us. When we have time to think, we can rationalize reasons to not care, to not intervene, to not be righteous. Or we can start thinking of ways in which to try to trumpet our own righteousness. But that's not the point of righteousness.
Jesus' concern is simple. He comes to win you salvation. Why? Because it's just the right thing to do, when you see someone trapped and dying in sin, to try to rescue them. Jesus is righteous, and this is good for you.