I remember playing Warcraft 3 but I do not remember a scene with Arthas' death. In contrast the lore tells us that every Death Knight died before they become what they are. So did Arthas die before he became one?
No, in Warcraft universe they are three generation of death knights.
First generation (or DK of Old Horde) were made by Gul'dan of slain warlocks of Shadow Council. Souls were merged by fallen warriors of Stormwind. This DK were more like necromancers - spellcasters. The most known DK of first generation is Teron Gorefiend.
The second generation of DKs from Lich King itself. Arthas was the first DK of the second generation. Difference between first and second generation is that Lich King's DKs were brute warriors and not so much spellcasters. The key to become DK was to give your free will (soul) and not your life. In War 3 expansion, after Frozen Throne was damaged and Lich King was losing his power, Arthas was gaining back his human nature.
We also have a little plot hole about Frostmourne. The lore says that anyone who is killed by the sword, his soul will be trapped inside, yet Lady Sylvanas Windrunner, after she was raised from the dead, managed to get both her soul and free will back.
The best evidence that you don't have to die to become DK is in third generation of DK or Ebony DK. These DKs are playable and when you select any living race to be DK you get their attributes and abilities. If you create a human DK you'll get human abilities and not undead ones.
Source - http://www.wowpedia.org/Death_Knight
There is a quest added in WoW in Wrath of the Lich King which, among others, in fact do show Arthas was a living death knight. In short, during the battle against Illidan he realized his own hearth is his greatest weakness (he was bleeding after the duel), so he rips it off his chest and throws it in Naz'anak - The Forgotten Depths below Icecrown Citadel. In the quest when player finds out the hearth, it's still pulsing, so, we could conclude that it also was pulsing when it was in Arthas chest (and along with bleeding), I would say this is a strong evidence Arthas was in fact alive when he turned into Death Knight.
Arthas hearth - http://www.wowwiki.com/Pulsing_Crystal
However, I'm not anymore so sure about third generation of DKs are they living or undead. In the very intro of the class, it is said player did lose his life. In fact, some people on offical forums and fan site are stating these DKs are brought back to life (reanimated). Also, the most of the RP players in WoW consider DK as undead. However, there are also in-game material which are stating otherwise. For example, flavor text in the DK talent Lichborn says - "Draw upon unholy energy to BECOME undead for 10 sec." So all in all, if we are take everything into account - lore, cinematics, gameplay, apperance, mehanics we could say that they aren't either alive nor undead, but something in-between (and have the ability to step from one side to another).
Just for clarification
From a 2007 blizzcon transcript with Chris Metzen and Alex Afrasiabi.
As far as I can remember, there is no specific scene shown, in where he dies. It seems that he more and more turns into a Death Knight, maybe in this transformation, death has occured somewhere, but without any noticeable changes or events.
From the wowwiki entry of Arthas Menethil
So one could argue that when the spirits fused that the old Arthas died. But it's a special case in my opinion since he merges with the Lich King and is no ordinary death knight.
So he didn't die due to a wound or any other injury.