Ok, i'll give this a go. To answer this, i'll need to go over some of the reasons why the war started in first place.

The Tsar of Russia, Nicholas I, was the head of the Eastern Orthodox Church and he was convinced the Ottoman empire was about to implode. The Tsar wanted to protect the Orthodox christians inside the crumbling empire. Constantinople (Istanbul) was the location of the principal cathedral (St. George's) of the Eastern Orthodox church. Which was, at the time, deep in the Ottoman Empire.

The French lead by Napoleon III, an ardent Roman Catholic, was protecting the rights of the Catholic Church and coveted the keys to the cathedral and like the Tsar, saw himself as the rightful leader of the christian population of the Holy land and sent the Marquis de La Valette to enforce his sovereignty and the Ottomans initially agreed.

However, in doing so, they had broken the Treaty of Kucuk Kaynarca with the Russians. As a result, it gave the Tsar the excuse he was looking for and he deployed his troops into the Danubian principalities. Attempting to force the Ottomans into reinforcing the treaty.

Which it did, as the Ottomans reversed the decision.

The French, in response sent the ship Charlemagne to the Black Sea. And Britain, sent a fleet to support the Ottomans as they did not want Tsarist Russia getting a foothold in the Mediterranean, via the Dardanelles and was concerned with the southern expansion of the Russian Empire getting too close to India

Both moves forced the Ottomans to change their decision yet again(!) and negotiations started as the Russians, French and the British would all lose face if they withdrew.

The differences of the churches was actually resolved at the table, but during the negotiations, the Porte of the Ottoman Empire received assurances from the French and the British that it would side the Ottomans if war was declared.

Negotiatians collapsed as neither the Tsar, Napoleon III or the British would back down and withdraw. Knowing it had Britain and France on its side the Ottoman Empire declared war.

So, that is how the British got dragged into it. So to answer your questions. Britains strategic interests, were maintaining the status quo and protecting its Empire and hegemony of the mediterranean, by propping up the Ottoman Empire. Despite its name, the Crimean war was also fought in the Baltic and the Pacific. And lastly, yes, Britain was pulled in by an alliance commitment to the Ottomans.

Source: Crimea, The Great Crimean War 1854-1856, Trevor Royle.