Churchill's premise was that, for a variety of reasons, the Western front had stalemated. The only prospect for advance was likely to involve high death tolls for minimal gains, and the road to Gallipoli stemmed from these words: "Are there not alternatives than sending our armies to chew barbed wire in Flanders?"

So, why Gallipoli? Well, as Churchill saw it, there were two alternatives. The first was the more dangerous. "The invasion of Schleswig-Holstein from the seas would at once threaten the Kiel Canal and enable Denmark to join us. The accession of Denmark would thrown open the Baltic. British naval command of the Baltic would enable the Russian armies to be landed within 90 miles of Berlin." The initial objective would be the island of Borkum, a few miles north of the Dutch frontier.

The alternative was to force a passage through the Dardanelles, and with occupation of the Gallipoli peninsula, insert a fleet into the Sea of Marmora, which could then advance to Constantinople, threaten the Turkish and force them to sue for peace, and bringing Greece, Bulgaria and Romania onto the side of the Allies. They could then continue to keep supplying the Russian forces as well.

Gallipoli was chosen over the Baltic approach largely due to a dispute in the Admiralty between Churchill and Lord Fisher. I don't profess to have much knowledge over the personal intricacies of Lord Fisher, but Churchill supported the Aegean proposals, whereas Fisher tended to focus on the North Sea plan, which had become almost his sole focus after the twentieth-century creation of the German high seas fleet. The pair were antipodes, and as Jenkins puts it: "It was the classic example of putting two scorpions in a bottle, with the added complication that in the Fisher-Churchill case they not only had the capacity to inflict most venomous damage upon each, but also had an intensity of emotional relationship, more appropriate to a love affair than to the stress of a great war."

Churchill won this tumultuous battle, as we know, perhaps because of Fisher's age. He was 74, and worked incredibly long hours, to the extent that he was once humiliated by the soon to be First Secretary of the Cabinet who came into his room in the middle of one morning and found him asleep. Ultimately, Fisher could not withstand Churchill in argument, and acquiesced, when he did not agree.

The critical weakness of the plan itself was the failure to plan for an integrated naval and military operation from the outset. Thus, the ships advanced down the strait, with minesweepers ahead in order to clear the mines laid by the Ottomans, under heavy artillery fire. When this failed, due to obsolete battleships (as the Admiralty had predicted heavy casualties so did not send her best, naturally), and misinformation about Ottoman ammunition, only then was the military sent in to clear out the artillery. The landing on the beaches itself is fairly complicated, but the key point is that once the Allies landed, they did little to exploit the situation, and apart from a few limited advances inland by small groups of men, most troops stayed on or close to the beaches. The Allied attack lost momentum and the Ottomans had time to bring up reinforcements and rally the small number of defending troops. Since the majority of Ottoman troops were inland, in order that they could relocate quickly, this was particularly disastrous. Troops from the UK, Australia, France and New Zealand then got bogged down in a long war of attrition, which after several offensives, ultimately lead to stalemate, but with terrible and worsening conditions for those involved. Both sides suffered, and summer heat and poor sanitation resulted in an explosion in the fly population. Eating became extremely difficult and unburied corpses became bloated and putrid. The precarious Allied bases were poorly situated, which caused supply and shelter problems. A dysentery epidemic spread through the Allied trenches at Anzac, while the Ottomans also suffered heavily from disease which resulted in many deaths.

At this point, Bulgaria entered the war, and this seriously undermined the position of the troops at Anzac, as the Germans could supply artillery via Bulgaria, and reinforcements were required for Greece, which slowed the supply of reinforcement to Anzac, and ultimately meant that the Allied position became untenable. I've tried to explain why Gallipoli was chosen, based off the background you asked about, and it is considered a failure.

Sources

Churchill - Roy Jenkins

Gallipoli - Les Carlyon (going off my memory of this book, so forgive any errors on the military side of things)

It was poorly organized.

Not only did the Mediterranean Expeditionary Force intelligence service lack sufficient intelligence, the Allies didn't even have any maps of the Dardanelles, instead relying on drawn pictures of the area taken from the sea.

As a consequence, the ANZAC forces landed a mile north than where they were supposed to land. When they landed they faced a steep slope that descended to the beach on three sides, so that it looked like a little amphitheatre. In order to not get caught out, they had to reach the top ridge before the Turks got there and this they failed to do. The Turks assembled in time to stop the ANZAC forces from reaching the ridge two and a half miles above ANZAC cove.

Source: The First World War by John Keegan.

I can't believe that no one has mentioned the critical role played by one Mustafa Kemal, aka the future Ataturk, during the Gallipoli campaign.

In "A World Undone," by G.J. Meyer, he is described as, "an eccentric young lieutenant colonel." Apparently, the rulers of the Ottoman Empire at the time disliked him so much that he was relegated to inactive status at the beginning of the war. He was also the commander of the only Turkish force within a day's march of Gallipoli.

Meyer goes on to argue that the British could have easily turned the campaign into a rout if they had seized the initiative at three undefended beaches during early hours of the landing.

In describing the ANZAC landing, Meyer notes they landed a mile or so north of Gaba Tepe, in terrain so harsh and confusing the Turks did not know it very well. Advanced elements of the ANZAC forces actually managed to seize the high ground at Chunuk Bair and Sari Bair, but shortly thereafter Ataturk arrived with a single battalion and cleared the ANZAC forces off the hillcrests.

After a lot of desperate back-and-forth battle the Turks eventually forced the ANZAC forces back down to their landing beach where British general Ian Hamilton (safely ensconced on a battleship offshore of course) ordered the ANZACs to dig in, not realizing that the Turks had spent their strength.

Meyer is not great with dates, and he condenses the Gallipoli campaign far too much. The Wikipedia article on Gallipoli is not half bad if you are looking for more detail without spending much money.

Overall, ANZAC forces acquitted themselves quite well despite mistakes in the landing location, poor intelligence, and below average performance by the campaign's commander. One example is the Third Attack at ANZAC Cove where ANZAC forces numbering 17k inflicted 13k casualties on Ottoman forces numbering 42k while only receiving 600 casualties of their own.

In October 1915, Ian Hamilton, the Gallipoli campaign commander, was recalled to London and sacked. If you look at his background, it could be argued that he was the wrong general for Gallipoli from the outset. Before WWI, he spent the years from 1903 to 1910 in largely administrative positions. You could argue that he was the Admiralty's scapegoat because he was not involved in the planning for Gallipoli.

Another mystifying advantage held by the British that does not seem to have been exploited was the the availability of the HMS Ark Royal and her seaplanes. The (few) accounts I've read do not make much mention of any attempts to exploit their aerial reconnaissance capabilities.

FYI, the latest episode of the AskHistorians Podcast dealt with this issue briefly (covering Anzac Day in a wider sense).

How accurate is the Gallipoli movie?