いっぺん means “one time”. You might know 一回 or 一度 which also means “one time”. They are interchangeable most of the time but いっぺん is more casual and colloquial. So you can also say 一回 or 一度死んでみる？ but with いっぺん it sounds like dying is a very casual and easy thing which makes her sound more cruel.
But there are differences between them. いっぺん and 一度 can also be used as emphasis. On the other hand, 一回 is simply used for number of times.
Also, いっぺん and 一度 followed by the particle に means “all at once” or “at the same time”. In this case, you don’t usually use 一回.
死んでみる means “try dying”. The grammar formation is te-form of the verb plus みる and it means “try doing something”. For example, 食べてみる (try eating), 飲んでみる(try drinking), 聞いてみる(try listening), 見てみる(try looking), 読んでみる(try reading), 書いてみる(try writing). Please note that this みる always written in hiragana is different to the verb 見る meaning “to look”. Also, this phrase in a question is used for making suggestions.