First day
Go find ALL the chests you can; do ZERO building during this time. Do not adhere to any specific class until you find the Wand of Sparking (brown surface chests), Snowball Cannon (ice chests), Slime Staff (extremely rare, 1/10,000 from any slime), or Ice Blade (ice chests).
If you found the Wand of Sparking:
Keep ALL the cobwebs you break. Try to have a large supply of these webs stored somewhere, as well as 10 of any gemstone, preferrably emerald, ruby, or diamond. Keep ALL mana potions you find, but restoration potions are the best variant here if you are lucky enough to get any this early. You will want to get 27 Iron/Lead Ore, a Furnace, and some wood. Turn your Iron/Lead Ore into bars, then craft an anvil. With this anvil, use 1 Lead Bar to create 10 chains, costing 1 bar. Now, create a Sawmill using 1 Chain, 2 Lead/Iron Bars, and 10 Wood. Next, create a Loom with several wood (~20 should do the trick). Craft whatever gemstone robe you can, the higher quality the better. Now, go hunt for more Emerald/Ruby/Diamond, because you are going to make the appropriate staff. A Ruby/Emerald Staff will do nicely for the time being, but a Diamond Staff is what you want to really go for. After you have the appropriate staff, you'll want to hunt for Tim. He will be significantly easier to find since his spawn rate goes from 1/200 to 1/50 because you are wearing a robe. After you get the Wizard Hat, you are almost ready. Next, you'll want to get a Nature's Gift from the Underground Jungle, but be careful, since almost everything can kill you at this point. While you are in the Underground Jungle, try to get enough Jungle Spores, Vines, and Stingers to create the Jungle Leggings, giving you some half decent defense and increased mana, reduced mana usage, and increased magic damage. The set bonus will activate from the robe and the hat, but any leggings will not alter the set bonus, so it allows for Molten Greaves later on. Now, return to the surface with your Wizard Hat, Gemstone robe, and Jungle Leggings. Go back near your spawn and build 3-4 rows of 125 platforms each, and place campfires on each end and in the middle. Then, using a Suspicious Looking Eye you probably found while underground, summon the Eye of Chtulhu and just run in loops around the arena, hitting J to heal your mana when need be and just spam your staff at him, but aim for the middle of his eye so you kill the servants when they spawn. Use the dropped hearts to gain extra health mid/late-fight; don't pick them up until he has transformed into his second phase, when you will need them the most.
If you found the Ice Blade:
Go mining until you get full platinum/gold armor, and build the same arena mentioned for the Wand of Sparking. Any accessories that improve defense need to be equipped at all times, and farming for a Warding/Armored Shackle is advised. If you can, gather Daybloom and Mushrooms, get some Water Bottles and Iron/Lead Ore, and make some Regeneration Potions and Ironskin potions. These will give you enough power to survive the Eye of Chtulhu's attacks and, most importantly, not die. The fight will probably last ~3/4 of the night, but if you have full warding/armored accessories, and full Platinum/Gold armor, it should be a cakewalk, especially if your Ice Blade is Legendary or similar.
If you got the Slime Staff:
It's not going to help you here, so I recommend just going to find the Snowball Cannon/Ice Blade. But if you insist on going full-on summoner, then get the merchant to move in and buy a ton of shurikens. Make the arena as described for Wand of Sparking. Try to get accessories that have Menacing/Angry or Armored/Warding, the extra damage for your baby slime and the extra defense for your own longevity. Run back and forth so the Eye will constantly chase after you in a horizontal motion, allowing your baby slime to basically stay in his hitbox about a half of the fight. The shurikens are for when he is floating above you or in his second stage, you just keep throwing them until you run out. Also, use the shurikens to take out his servants, as they can amass much faster in Expert.
If you found the Snowball Cannon:
Try to get full Platinum/Gold armor and try to get a lot of snowballs, as this weapon will last you well after the Eye of Chtulhu if used right. If you can get armored/warding accessories, this will work; what you really want are Hermes Boots with some kind of damage increasing prefix. Another accessory to go for is the Anklet of the Wind. Build 2 big rows of platforms about 300-400 platforms long, spacing out campfires along the way. Just run back and forth firing snowballs at him, but aim for his servants in particular, since they will drop hearts. The fight should last no longer than ~2/3 of the night if you are good at dodging and aiming.