If you have 4 Likert items, for example y1, y2, y3 and y4. Then a usual way it to just sum them, like:
Index1 = y1 +y2+ y3 + y4.
Or you can,if you prefer that, you can give different weights to different items (like giving higher weight w1 to y1 if you think that y1 is more important than the rest).
Index2 = w1*y1 +w2*y2+ w3*y3 + w4*y4,
where w1+w2+w3+w4 = 1 (so that the weights sum to 1, just like the weighted mean.)
Now you can use Index1 or Index2 to calculate a correlation coefficient towards an other variable.