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Premises: $p\implies m$, $m\implies t$, $m$.

Conclusion: $m\implies p$

My goal is to provide a counter example for the following problem since it is not true. I am familiar with writing counterexamples but on more concrete proofs.

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This can be attacked directly: how can $m \implies p$ be false?

You may find it helpful to covert the implication into a disjunction (that is, express $m \implies p$ using $\lor$ instead of $\implies$) and then negate it.

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    +1, but $\vee$ is called disjunction, not conjunction. – 2017-01-24
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    @AlexKruckman Right. I have corrected my error. – 2017-01-24
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There are three variables here; each can take two different values (true or false). That means there are only eight different situations to consider. That's small enough to do the straightforward thing: check 'em all, and see if any are counterexamples.

For example, if $p$, $m$, and $t$ are all "true", then $p \implies m$ is true, $m \implies t$ is true, and $m$ is true; so the premises are true. $m \implies p$ is also true, so this is not a counterexample.

For another example, if $p$ and $t$ are true but $m$ is false, then $p \implies m$ is false, so the premises are not all true; that means that this isn't a counterexample either.

There are six more cases to check - I'll leave those to you.