When do assists show up in the kill feed?

The new assist visual that shows up as of the new patches are for assists only, being a part of the kill (doing damage as you say you are) doesn't count as an assist, but rather an elimination. If you've done damage to an enemy recently and someone else finishes them off you get credit towards the elimination as an elimination, not an assist.

From the late May update:

• Your final blows are now highlighted by a white background in the Kill
• Feed Added new Kill Feed icons for environmental kills and a number of abilities
• Heroes resurrected by Mercy are now are now highlighted and color-coded in the Kill Feed
• Certain types of assists, like McCree’s Flashbang or Zenyatta’s Orb of Discord, now show up in the Kill Feed

Ways to get assists are through means such as recon via Hanzo or Widowmaker, or offensive/defensive assists via Zenyatta's orbs. Other ways include healing/buffing from Mercy or sleep darts from Ana.