There is no one way to "properly" speak Shakespeare's verse lines. Throughout the history of Shakespearean performance, actors have often approached the text through highly individual techniques. This requires experience in order to develop a personal methodology that works for you.
Shakespeare himself most often chose a form called iambic pentameter when he wrote his lines in a structured verse (as opposed to when he wrote lines to be delivered in ordinary prose, as with Falstaff). Iambic pentameter simply means that each line has ten metrical "feet" or beats before your eyes move to the next line. The most typical rhythm to this structure is "de-dum de-dum de-dum de-dum de-dum", with breaks in this rhythm to indicate changes in the character's thinking or feeling.
A good way to speak Shakespeare's verse for the first time is to choose a line and to "force" it to this metrical requirement even if that feels awkward or embarrassing at first. Often, the forcing of this beat makes an actor aware of its necessity or its deviation, and the reason for such changes as required by Shakespeare. After making yourself aware of this structure, you can speak the lines with this rhythm subtly underpinning your speech.
It has often been commented that this rhythm is like a heartbeat, lying underneath the words and propelling it along. This can feel true when extensive experience makes it more natural to speak in such a rhythm. But you may also notice that, just as often, the rhythm to speaking Shakespeare's verse is more like the syncopation of jazz music (notice that the strong emphasis is on the second beat, just as in syncopation). This choice in rhythm makes the text skip along rather than plod along, much like a musicologist would tell you can be the case in some rock music as opposed to jazz.
It has often been noted by theatre practitioners such as John Barton that everyday colloquial conversation often naturally assumes an iambic rhythm so that to speak Shakespeare's verse is to speak as we often do in everyday life. This can often be true of the experienced Shakespearean performer, who can try to speak Shakespeare's verse as if it were the naturally occurring thought process of a contemporary playwright.
The wonderful aspect of Shakespeare's verse is that the more you analyze it, the more interesting it becomes to speak it. Let us take the example of perhaps the most famous verse line in all of Shakespeare’s plays: “To BE or NOT to BE, that IS the QUEStion”. This famous line of Hamlet’s breaks a basic rule of iambic pentameter - it is eleven feet, not ten. This is called a feminine ending, because it ends not with the expected strong beat, but an unemphasized beat. If we are following the basic idea of syncopation, this would require us to move directly to the next strong beat rather than to rest on the weak ending of “QUEStion”.
The next line begins with “WHEther ‘tis NObler”, and we are back onto our regular rhythm for a time. And yet, the most famous line in Shakespeare is often spoken mistakenly as a statement, where the actor ends after “question” and pauses before moving onto Hamlet’s rationale of that statement. An astute observer of those lines will notice that this requires the speech to keep moving along, rather than to break down into over-reverential parsing of lines as if they were rich dishes.
This brings us to the key observation about how you should speak Shakespeare’s verse: learn its meaning, observe its rhythm, but never speak it as if it were some of the most recited and quoted lines in world literature. This is of course one of the hardest things to accomplish when speaking Shakespeare’s verse, but if you can take those words and speak them with confidence and purpose, ignoring the pitfalls of making them set pieces and treat them as the fabric of a larger tapestry, then you are well on your way to speaking Shakespeare's verse as if it were your own thoughts and words, and by modern acting standards, that is indeed the question to be answered.