It’s entirely possible to work the problem using only ordinary English. However, you can also approach it systematically. The statement of the problem involves four propositions. In the order that they appear they are:
$p$: John committed the murder.
$q$: John was in the victim’s apartment.
$r$: John did not leave before $11$.
$s$: The doorman saw him.
Here I’ve given them symbolic names for brevity. Now translate the assertions in the three sentences into statements in propositional logic:
\lnot &r\to\big(s\land\lnot(s\lor p)\big)
If you do a little ‘algebraic’ manipulation of the third line, you may be able to see right away whether the system of statements is consistent. If not, write out a complete truth table; since you have four basic propositions, your truth table will have $2^4=16$ lines. Is there any line in which all three statements are true? If so, they’re consistent; if not, they’re inconsistent. (Notice that you really only have to construct the half of the truth table in which $q$ is true, since you don’t care about the other eight cases.)