If you're choosing to sanitize to have a more dependable brew, start by sanitizing everything that will come in contact with your mead.  This includes the glass jug, funnel, spoons, spatula, rubber stopper and water lock.

Pour the yeast packet into a cup of water to allow the yeast to rehydrate and wake up for at least 10 minutes.  Yeast don't function as well if they're just dumped from a dormant state into a glut of sugar.  You have to ease them into it.

Place your blackcurrants in a saucepan and just cover with water.  Bring the water to a boil and then turn off the pot.  Allow them to sit for 5-10 minutes in the just boiled water so that the juice can extract.  Pour the currants through a sieve, extracting the water and juice.

Wash the remaining currant pulp in more water to pull out the rest of the juice.

Place all your extracted currant juice into a saucepan and add 1 quart of honey.  Warm the liquid until the honey just dissolves, but do not boil.  Allow it to cool a bit so you can safely handle it, and then pour it into your glass jug.

Once the honey/juice/water mixture is between 90 and 100 degrees, add in your hydrated yeast.  If necessary, add more water to bring the brew to within 2 inches of the top of the jug.

Put on the rubber stopper and the water lock, making sure you add water to the water lock.

Within 24 hours you should see bubbles and active fermentation.

Allow the brew to ferment until active fermentation has stopped, roughly 4 to 8 weeks depending on temperature.

At that point, you're ready to bottle your mead.  You can get a wine bottling setup, or use the simpler method and pour it into reusable grolsch bottles.  Mead tastes best if allowed to bottle condition for at least 2 weeks, but preferably 2 months or more.  I'm still enjoying bottles from 8 years ago, and each one is better than the last.

Enjoy!