Well, first of all and as you seem to well know: Not dying is more important than getting kills. But getting kills is a matter of knowing what champions you want to play - for someone like LeBlanc or Pantheon, you'll be getting kills by blowing enemies down to two-thirds or half health in harass and then just exploding them with your burst. For Ryze, you're looking for long trades that favor your spammable Q, and same with autoattacks for AD carries. But some tips:
- The best way to get kills is to be good at slowly lowering your enemy's health in lane. Then, when they've fallen low enough you can jump on them for the last third or so. If you're playing someone who doesn't do particularly high damage (Ashe, say) you need to slowly get them low enough until you can flat-out win.
- Towers aren't invincible defenders. Especially if you have a tank you can, if you know you will get the kill, dive their tower. Try to keep the tower aggro on the tank by having him hit the enemy first, or trade tower aggro when you get good enough to manipulate it well.
- Stealing kills is going to happen, unfortunately. Don't worry too much abaout it.
- Getting killed is a matter of knowing when and when you can't win fights, you'll get the hang of it as you play more. Some champs are extraordinarily hard to duel (Jax, Xin Zhao) and some are very slippery and will kill you if you try and get the finishing blow and get worn down / their cooldowns come back up (Singed, Fizz).
Also, if you're playing a supporty tank or support mage like Nautilus, Alistar, Maokai, Soraka, or Janna you probably won't be getting many kills - you just don't do enough damage. Maybe early-game or midgame with ult on Nautilus, but generally your giant fight and lane impact doesn't rely on damage so don't worry about it. 0/2/25 is just as impressive as 8/2/4,and 0/0/30 is always a great one.