Based on a Lavos Skeleton and fleshed out by me
Exeggutor is one of the most versatile status spreaders in the tier. With a great movepool, it can sleep, paralyze, and has the option of booming when it's job is done or finds itself in dire straits all the while being fairly tanky. With all these assets, it has carved itself a place on most offensive teams in the tier, especially boom offense.
Defensively, Exeggutor can serve as a secondary check to threatening sweepers like Vaporeon (if you're packing Giga Drain), Marowak and Machamp, just beware of any unexpected Ice Beam from Vapo and Hidden Power Bug or Fire Blast on the switch-in from the latter two.
Set 1: Standard Explosion
Move 1: Sleep Powder / Stun Spore
Move 2: Psychic
Move 3: Giga Drain / Hidden Power Fire
Move 4: Explosion
Pick your status of choice (usually sleep, but you can opt for paralysis if your team already has a sleep user or benefits more from paralysed pokemons) so you can start annoying your opponent or at least cripple the things you can't hit well. Psychic is your main attacking move, since it threatens near-3HKOs on the electrics and the random SpDef drops will most likely force a switch from your opponent. Giga Drain provides safety against Tyranitar, healing off Crunch/Fire Blast damage, effectively avoiding the 2HKO, and gives you better matchups against water types even though it has low pp. On the other hand, Hidden Power Fire destroys Forretress and dents Skarmory (who otherwise outright wall the Giga Drain set) at the cost of less bulk due to DV requirements for the move. Keep in mind that unless you have done prior damage or hit Skarmory on the switch, it still wins the 1v1 due to being faster and having Drill Peck to hit you Super Effectively. Rounding up the set is Explosion, that, coming off from a base 95 Atk, can secure OHKOs on most non-resists in the tier after some chip damage (the best non-resist being suicune, which needs at least 85% health to survive more than half of the time), making switching into Exeggutor quite risky at any time in the game.
The utility of paralyzing, sleeping or booming common switch-ins such as electrics, snorlax and skarmory can mean opening up a path for another teamate to sweep, so always be on the lookout for opportunities that exeggutor can create. Some examples are booming on Raikou and Zapdos so your Vaporeon can set up growth more safely or sleeping/paralyzing skarmory so belly drum snorlax has a much better chance at taking it out after setting up.
Set 2: Leech Seed
Move 1: Stun Spore
Move 2: Leech Seed
Move 3: Psychic
Move 4: Substitute / Explosion
This set can be a one man army against many teams coupled with Spikes support. Leech Seed activates in GSC when the seeded pokemon attacks, meaning it forces switches almost every time it lands. Stun Spore, coupled with Psychic's raw power and SpDef drop chance, makes it even more difficult to stay in and take the Eggs head on, forcing switches and racking up residual damage. Substitute increases Egg’s longevity and prevents it from being hampered by status users, but Explosion might catch many opponents off guard while cycling through switches trying to run you out of leech seed pp.
Anything with Toxic makes a good partner for this set, since it has the potential of racking up a ton of residual damage. The fact that this might force some key defensive threats to use Rest might help a teamate's sweeping chances, especially considering you can still keep the option of nuking any threat with Explosion.
A 4-attacks set is viable, where the status move is foregone in favor of maximum coverage, but this should only be done when your team already has a sleeper and you don’t benefit much from spreading para with stun spore. Exeggutor is also a decent Thief user, as it commonly baits in many threats like electrics, skarmory and tyranitar, removing their leftovers, which can then be capitalized upon by racking up spikes and/or chip damage so a teamate sweeper has an easier time. Just make sure Exeggutor has no items when picking this set, of course.
If you'd rather spread as much status as possible, you can forego your coverage move on the standard set and go double status so you can nab a surprise sleep or paralysis on your opponent after revealing one of the two and they assume you're running the Standard set, but keeping the coverage is usually the better option as there are many pokemons who can switch in to double status Exeggutor that don't mind being statused as much, most notably Rest+STalkers such as Snorlax, Suicune and the electrics.
[Checks and Counters]
Most prominent Special walls in the tier can switch safely enough into Sleep Powder and abuse Rest + Sleep Talk loops to whittle down Exeggutor with their moves like the Electrics, ST Snorlax and most bulky water types if Exeggutor's not packing Giga Drain who have a way of dealing damage to Exeggutor like Ice Beam Suicune. Do keep in mind that almost all of them fear Explosion, so beware of possible mind games such as booms on the switch. Baiting the Explosion is also possible, of course, and most normal resists such as Skarmory or Tyranitar can take one rather well. Other notable Explosion mitigators include pokemons with Reflect (Raikou, Zapdos and, to a lesser extent, Dragonite) and Charm Umbreon. Umbreon actually takes low damage from Egg’s other attacks, can outspeed it and has pursuit, even if it's a rather weak one. The other better pursuiters in the tier are Tyranitar and Houndoom, who can also improve the mindgames if they have Crunch. Both fear statuses (Rest Tyranitar not so much though, but it usually doesn't have Pursuit and Rest in the same set) while Houndoom fears booms and Tyranitar fears Giga Drain. If you're facing a set without Hidden Power Fire, most Steel types can switch in and threaten Exeggutor even though they must beware of status.
EDIT: Made the changes mentioned bellow
EDIT 2: Removed Synthesis on the OO as Lavos suggested