Here's a screenshot of his Washington vs Siamese game:
I'm not going to play it out (its on epic, kind of slow), looks like a hard fight but I think its winnable.
Don't fortify borders while on plains. The -33% plains penalty kind of negates the fortify, and since he has superior troops (longswordsmans) those pikemen would die if he sneak attacked.
You're kind of in a bad situation because your economy sucks. Two of your cities have 1 pop and your cities really don't have many buildings (other than your capital). Did you concentrate on your military early game? Did you fortify the border first or did he? I think you waited too long while he was building up an military, economical, and technological advantage.
Since you're playing in prince setting, the game would have been relatively even at one point. It looked like he expanded quicker than you; you should have taken advantage with an preemptive strike.
Edit: since my play style usually involves a small military, I was actually surprised to see that many enemy military units. Does the AI build up their military whenever they see units on their borders? A lot of my conflicts are decided by 5 units (3 infantry/horsemen, 2 artillery). After I destroy the first 10 units without losing any of mine, the AI usually surrenders with a favorable peace treaty.
I won't be playing this game out either and I play on Emperor and normal speed. I presume the only speed difference is that my games are 500 turns and this is 750 turns? Anyhow, I will assume that we can make comparisons based on the date and this is 1604 AD. Since I have a slow day here and like helping newer players (returning the favor for the help I've taken from these forums and Sulla's site after having returned to Civ after nearly a 15 year break from the game) this will be a long post.
First of all, you're in an extremely bad situation here since you have very little research, a bad gold situation, and -4
with little chance to get
until those col's are built which seems rather far off. Your lack of research (and also econ) are likely due to lack of city growth since your overall pop. seems quite low. Fixing this economic mess into an economic and research powerhouse will be next to impossible, but we can improve it and I'll give you advice as to what you should have done earlier.
The first thing you need to do for econ is to get happy and stay happy while growing some more pop size. This means that currently you need those col's. When unhappy, and most of the time once my cities have grown to several pop and I am planning to soon aquire more cities, either by REX/ICS or conquest I put all of my cities on Production focus as long as no city is starving. If you try that for Boston and Phila, you may find that you get those col's sooner. I am unsure what New York (I assume that city in the far south) is doing but if it can get a work boat to Atlanta at some point you can work those fish for a nice 3G, 3F tile. Anyhow, when unhappy, cities should usually go to production, or something other than default..perhaps gold or science fcous.
I presume that unit maintenance is eating not only your dollars but your coins as well. How do we rectify that? Well, we can delete some units, but by far the better way will be to take a bunch of Siam's units to the grave with some of our brave troops. Siam has you enveloped here and it might even get worse if he builds another city or two to your south and/or allies with those two CS down there. Siam has more pop than you and clearly will continue to outtech you if you leave the status quo. You share so much common border with Siam and have so many troops massed vs each other that war seems inevitable.
So, from the above we see that you must prepare to DOW Siam ASAP and noting that you need some of his space for your own growth and some of his cities will make nice puppets. I am not much of a pike user, however, I presume that they really can hurt elephants. I do know that elephants can really hurt horses so be careful there. Looking at the screen shot we see that Siam has 3 or more likely 4 exposed elephants and you have lots of pikes handy. Can you prepare this turn to hit each of those phants with two pikes next turn when you DOW? If you can kill those 4 phants on the turn when you DOW (2 near Atlanta and 2 near Chicago) you'll put a big dent into his army. Of course some pikes will die in the counter attack, but fall back into your territory and use the ranged units to snipe at his counterattacking advance..garrison them in border cities if needed rather than letting them get killed. Horses can pick off wouded unit and then run back out of harm's way.
If you can waste much of Siam's army here your game is tenable, I think. Then you conquer some of his cities, in a clockwise manner, razing Luang Prabang, and puppeting Martaban (notice that lux for 5
)..perhaps then you can continue into Kamphaeng Phet, but you'll likely need to heal and may need reinforcements and perhaps Siam will beg for peace and give you stuff and you can decide which route to prefer.
Are your workers on automated? If so they have made a mess of things. I NEVER put my workers on automated. For me, having my workers do what I want to optimize my empire's area plays a major roll in whether I'd struggle on king level with low gold and productivity or whether I stomp the AI's with ease on Emperor (OK we'll see if I can do that on Immortal next game).
Here's what I'd have done to better use your small empire's resources.
1) Your road layout is poor. There are many ways to use just 6 tiles of road to connect Washington, Boston and Phila. Then you could make a road use just as many tiles as you did to NY in the south. I'd make that road on the desert tiles since once my worker finishes a road on one tile, he can move to the next and start the road right away. Unless NY has some decent pop, you may be losing more gold in resources than trade income that connection provides and while the movement ease is nice, there's no real attack threat to NY here yet.
1a) Speaking of roads, the Meritocracy SP is very nice since it provides +1
for each city connected by road and here that would mean your empire is not
.
2) I see two CS in the south. I'd suspect that with both being coastal, one of them is Maritime. You should strive to be allied with that Maritime CS once you start building a couple new cities since you'll be fed and can grow to a pop of several in each with ease. Then you can build trading posts on any tiles except those river tiles (usually grassland) that will benefit from civil service (noting your pike style).
3) There are some nice river hills to the south of Washington. TP's on them will mean 3G and 2P from each of those tiles. I also see too many farms on tiles that only yield 3 food. With a Maritime ally, those should have TP's on them.
4) Probably you shouldn't have build Chicago and/or Atlanta when that were driving your empire into unhappiness.
5) Once you get some experience with Maritime CS ally(s) and more TP and learn to better manage and improve empire growth, you'll be soon breezing thru King level.
OK to recap and stressing that you'll learn a lot here trying to save this game, but you made need to go back 100 turns and do some things better (that will also be a learning experience to try).
1) All those cities go to whatever focus speeds up production (unless it starves them) until you're clearly happy.
2) The 2 pikes near Atlanta are fine as they can step and then strike that phant in Maraban. The two pikes north of Atlanta are fine as they can stab that pahyderm to death. In the map center just south of the wall, move that pike and sword north next to that phant. Do the same with the two pikes in Chicago. Then on the following trun DOW Siam and try to kill 4 phants. Your catapult in that citadel (I think) can likely take fire from that city but use it to pound enemy troops and retreat it if you think you'll lose it. The next GG you get can give you that nice 25% bonus where needed in battle.
3) You may lose Chicago in this war, but so what, it is a 1 pop city that can't grow due to
and losing it will reduce unhappiness. Don't lose Atlanta, since then Siam will be too close to your empire's heart. Speaking of Atlanta, those wall won't be built in time to help this game since either you'll have rolled up a couple Siamese cities along that NW coast or be overwhelmed by his counterattack. I'd switch production in Atlanta.
4) Once you're happy and have taken a couple Siamese cities, grow and stay along the edge of happiness and tech and you may yet end up OK.
.. neilkaz ..
Unhappiness from number of Cities reduced by 50%.
for each city you own 
connected to the 
Capital.
Citizens in non-occupied Cities by 20%.
/turn. Siam is willing to buy Chicago for 140 
