Finding the Formula
This article will not attempt to articulate the finer points of Ryu's offensive game. It is up to each individual player to understand how best to apply the tools they are given. There are however, a set of core attributes that the writer believes are common among accomplished Ryu players. They are as follows:
Solid anti-air capabilities
Ryu's ability to control the vertical space directly above him is 50% of his game. Your opponent should not be able to jump at you, period. This principal holds especially true in the Ryu mirror. Against opponents that are able to change their jump trajectory mid-air, substitute Ryu's Dragon Punch for cr.HP. Good Ryu players DP every jump-in, every time. The secret to ensuring consistency lies in learning the DP shortcut; it is as follows (when facing right): down-back, down-forward, down-back, down-forward + MP. The entire move is executed from the crouching position. This has the additional advantage of lowering Ryu's hitbox and thereby giving the player more frames in which to execute. Players who experience difficulty in being able to DP consistently can use Ryu's standing roundhouse (st.HK) as an initial substitute to become familiar with the timing.
EDIT: Thankfully, Ryu's dragon punch has received a significant buff in v2012. Players should remain using the DP shortcut listed above; Ryu's fierce DP however is now only single-hit, with damage totalling 160 during the first and second active frames only.
Consistent blocking skills
As mentioned earlier in this article, Ryu players will often have to endure a torrent of abuse from more able-bodied members of the cast while waiting for an opening to strike. As Ryu is not C. Viper, solid blocking skills are essential. Knowing when and how to block is a grossly underrated skill in the game. When in doubt, just block. The worst that can happen is you will get thrown; a small penalty compared to eating a lengthy combo.
Additional note: At higher levels of play, blocking correctly sometimes includes not pressing any buttons. Fishing for counter-hits during a block string is a huge part of high-level play. Of course, the decision whether to block whilst OS-teching (at the risk of a counter-hit setup), or to simply block with no button presses is up to the player. The very best players apply OS-tech sparingly. For those unfamiliar with the concept of OS-teching, a complete explanation can be found here, courtesy of VesperArcade.
A solid ground-game
Ryu's single most useful and intimidating tactic on the ground is his bread n' butter cr.mk > hadouken. New Ryu players must develop the ability (and the confidence) to walk up > cr.mk against their opponent at will. The purpose of this is two-fold; first, it applies pressure on the ground and pushes your opponent back, secondly, it shows your opponent that you're not scared and are willing to take control of the match. Used in tandem with solid-anti air, one can effectively 'bulldoze' the opponent by walking forward and applying cr.mk > hadouken consistently and applying a DP when the opponent finally jumps. Be aware that better players will begin to look for the hadouken after the cr.mk and will often try to focus through it (or even ultra), so either apply the hadouken sparingly or be ready to cancel into a tatsu/DP if required.
Successful Ryu players know the cr.mk range. That is, once their opponent crosses that invisible line, it becomes almost second-nature to immediately apply cr.mk > fireball. This tactic works exceptionally well as a 'keep-away' tool, and will keep most members of the cast at bay while also goading them into jumping. This is Ryu's most effective defensive tool and is almost universally feared by most players. Better players will look for it, so as mentioned above be prepared to substitute the hadouken for an armor-breaker if the opponent begins to get excited with focus-attacks or otherwise.
Ryu's cr.mp is a counter-hit machine. It has a large hitbox, is chainable and leaves you with decent frame-advantage on hit (especially on counter-hit). For more advanced players looking to create a counter-hit setup, this will be your primary tool. It is also useful as a counter against many characters' specials, and will stuff the following moves clean: Bison's bicycle kick, E.Honda's headbutt and Blanka's roll. These are just a few examples as there are many, many more.
An understanding of zoning
Ryu's zoning abilities represent the other half of his game. Effective zoning goes hand-in-hand with a solid anti-air game (see point 1). That being said, it is also perhaps the most difficult aspect of his game to master. The purpose of fireballs, first and foremost, is to make the opponent jump. Damage is secondary. However, one should take care not to throw a fireball while the opponent can jump at you. Rather, the purpose is to train your opponent to think you are going to throw, and apply anti-air when they attempt to jump it. Zoning boils down to a relatively simple 50/50 mind game (wait or throw?); the challenge lies in applying it consistently to many different types of opponents. More experienced players will not be as easily intimidated and will have all manner of counters ready (particularly non-shotos). Players should also note that there exists a number of characters in the game that are largely immune to Ryu's zoning game. These include C. Viper, Ibuki and Fei Long. Hence, these matches tend to be at least 4-6 in the opponent's favour. EDIT: This is likely to change in v2012, given Ryu's substantial buffs to his zoning game. Be aware of when to zone, and when you are being zoned. Be aware of characters that can be zoned more easily than others (E.Honda, Zangief, T.Hawk, other shotos).