Chapter 59.18 RCW (Residential landlord-tenant act) has the statutes for unlawful detainer (eviction) actions against tenants covered by the RLTA. Other statutes cover other types of rental. The statutes are at http://apps.leg.wa.gov/rcw/default.aspx?Cite=59 .
The general process is give the proper notice to the tenant, file an appropriate petition with an appropriate court after the notice period, provide proper notice to the tenant, have a court hearing, and carry out the orders of the court. The specific details as to all these events are very important. In general, a landlord's failure to follow the statutory procedure likely will result in dismissal of the landlord's action.
The costs depend on the complexity of the case. in the Seattle area, there are offices that offer a flat fee of $750 for eviction up to the initial hearing.
For a residential rental, a tenant can be removed from the premises by the sheriff in as little as 16 days. A practical time frame likely is closer to a month.
Just because a landlord wants to evict a tenant does not mean that the tenant will be evicted. The tenant may assert valid defenses and stop the eviction.
You likely should review your specific facts with your attorney for informed advice for you.