Flying from LAX-MEL next month in Business Class on VA.

Flying from LAX-MEL next month in Business Class on VA. I am also VA Platinum. Is there anywhere to have showers in the Tom Bradley Terminal at LAX prior to my flight?

Yup. You will have access to the new star alliance lounge. 

The first class section is pretty small, has some dinning options but nothing special.

The showers are out in the buisness class section never had to lineup for them. They were ok, got the job done.

As Jared said

You will have access to the new Star Alliance Lounge in TBIT

With you Virgin Australia Platinum Card you will have access to the First Class section and your also allowed to bring one guest (Virgin Australia is the only airline allowing someone to bring a guest into the Star Alliance First Class Section)

The Business Class area is bigger and thats where the showers are

Also worth checking out the terrace (just near the showers). 

Thansk Jared & henrus.  I'll be sure to check these all out!