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LingakuLingaku[verb て-form] + みせる [verb て-form] + みせます
てみせる goes after the て-form of a verb. About something you are going to do, it is a declaration – "I'll do it, just watch me"; about something done in front of someone, it keeps the plain sense of みせる見せる, "show how it's done".
The declaration is about your own action and stays in the non-past, often with ぜったいに絶対に, かならず必ず or こんど今度こそ. It has an edge of defiance or pride: you mean to prove someone wrong. The polite form is てみせます.
About someone else, the declaration only turns up when you quote their words.
In the past, or with someone else as the subject, it is usually the literal sense: doing something in front of someone so they can see how it's done.
The two look alike. てみる is trying something to see what happens; てみせる is doing it so someone sees.