There has been already many articles written about Alpha male. The classical "alpha male" is usually described as a leader of wolf's pack, meaning that the strongest one will become leader.
Which sounds great, but at the same time it is quite difficult in our society because we are not wolfs. Who is the strongest man, and in which area? D. Trump can be considered strong Alpha Male in business, but put him in fitness and he will fail to be even low Beta. Obama could be considered strong Alpha male in politics, but not having clue he would fail in business. Schwarzenegger could be considered Alpha in fitness, yet he would be a Beta in your local college among knowledgeable professors. So who is really "Alpha"? It's all relative...
There are also different types of Alphas. Is Obama loner? Is Trump loner? Probably not. How about Clint Eastwood, a guy who is being given as an example of great Alpha, is he a loner? He is always loner, at least in those movies...
So Alpha is sort of outdated term, it belongs to history. You can be alpha in one or more groups, but you may be beta in many different groups.
IMO better way is to shift away from this term and focus more on developing personal characteristics that are attractive to females. If you are not natural leader you will have hard time to become one, it takes too much energy and time. Does it even worth it? It may, but it really depends on what you want... Find something that you like, and try to become good at it. Being good at something will give you natural recognition from your peers, and mainly from girls, you will simply be perceived as sort of 'alpha' in that field...
There is lots of competition out there, if you want to become good at something you usually have to work hard. If you want to be a good weight lifter, you have to work hard in fitness. If you want to be a good martial artist, you have to train many many hours every week. If you want to run good and successful business, you have to work long hours, unless you find some unique idea or are in some way lucky. If you want to become good academically, you better study hard because even there is lots of competition. Competition is everywhere, the potential of becoming good without diligent work is pretty much minimal... That's the reality, so try to become good at something real by working on it, by investing in it - and it will be much better than faking some "alpha male" characteristics...
There is also another term, it is called Omega. It is usually guy who is a loner but at the same time he has lots of Alpha male characteristics as he is sort of rebellious. C. Eastwood probably fits description more of Omega rather than Alpha. Omega doesn't compete with anyone, he lives his own life on his own terms, he makes his own rules, he walks his own way, he does what he wants and could care less whether others like it or not. He has no needs to be a group leader, perhaps he even avoids it for the same reason you've mentioned - he doesn't want to deal with all the daily BS...
See, to become Alpha you have to spent lots of energy to get to a leading position. Say you are working for large company, you love your job, you work hard, but it will usually take you long time to claim up the ladder and be recognized as a leader of some group, perhaps become CEO. At the same time, you have to deal with lots of competition, there are always guys and girls who desire the same position... What more, there is usually somebody above you anyway, somebody who tells you what to do and how. So even though you eventually become an Alpha, does it really worth it? It does, you get the recognition, but it will cost you lots of extra energy, if you know what I mean. The reality also is, that once you become too big (meaning once you make good $$$), the company is usually seeking to replace you with somebody who is more productive for less salary...
Omega is little different. If he works for larger company he could care less about climbing the ladder. He may become a leader, however that's probably not his biggest ambition as he tries to avoid the work politic BS. Preferably he eventually leaves the company and opens his own business, where he is his own boss and doesn't listen to anyone. It might also be difficult, he might spent lots if time and energy, but he is his own boss, nobody tells him what to do, which carries lots of value...
There is also nothing wrong being half Alpha or half Omega. You simply work on your own, or you join a group of people with similar interests, you share, you become part of a team... As always, it all comes down to what you really want...
So, my two cents, stop worrying about being "alpha", it is pointless, useless. Work on fundamentals, work on being open minded, find something that you like, find your passion and become good in it. If you are working for company and like it, good; If not, open your own business if you can... That is one thousand times more valuable than being "alpha"...
There is not much differences in attraction between Alpha and Omega. Alpha gets laid (after all that hard work climbing the ladder) but so does Omega, especially if he develops good fundamentals. A good Omega gets actually laid much easier and more frequently than Alpha. Why? Well, see B. Clinton, Obama, Schwarzenegger, CEO's,... Society wants to crucify them should they have any affair, they have to be very careful with women, what they say and what they do... On the other hand, nobody really cares about Omega, he can do as he wishes, anytime, anywhere, without people even noticing...