In the guilt or innocence phase of the case (which is really not the one where a lawyer is likely to be the most effective in most cases like this one), the primary strategy is to force the prosecution to prove the case beyond a reasonable doubt and to point out at trial every way that the evidence fails to do so.
For example, Colorado Rule of Professional Conduct 3.1 (which has an almost identical counterpart in every U.S. state) says:
Rule 3.1. Meritorious Claims and Contentions
A lawyer shall not bring or defend a proceeding, or assert or
controvert an issue therein, unless there is a basis in law and fact
for doing so that is not frivolous, which includes a good faith
argument for an extension, modification or reversal of existing law.
A lawyer for the defendant in a criminal proceeding, or the respondent in a proceeding that could result in incarceration, may
nevertheless so defend the proceeding as to require that every element
of the case be established.
OFFICIAL COMMENTARY
1 The advocate has a duty to use legal procedure for the fullest
benefit of the client’s cause, but also a duty not to abuse legal
procedure. The law, both procedural and substantive, establishes the
limits within which an advocate may proceed. However, the law is not
always clear and never is static. Accordingly, in determining the
proper scope of advocacy, account must be taken of the law’s
ambiguities and potential for change.
[2] The filing of an action or defense or similar action taken for a
client is not frivolous merely because the facts have not first been
fully substantiated or because the lawyer expects to develop vital
evidence only by discovery. What is required of lawyers, however, is
that they inform themselves about the facts of their clients’ cases
and the applicable law and determine that they can make good faith
arguments in support of their clients’ positions. Such action is not
frivolous even though the lawyer believes that the client’s position
ultimately will not prevail. The action is frivolous, however, if the
lawyer is unable either to make a good faith argument on the merits of
the action taken or to support the action taken by a good faith
argument for an extension, modification or reversal of existing law.
[3] The lawyer’s obligations under this Rule are subordinate to
federal or state constitutional law that entitles a defendant in a
criminal matter to the assistance of counsel in presenting a claim or
contention that otherwise would be prohibited by this Rule. See A.L.L.
v. People ex rel. C.Z., 226 P.3d 1054, 1060 (Colo. 2010) (addressing
obligations of court-approved counsel for a respondent parent in a
termination of parental rights appeal).
Any evidence which tends to be exculpatory could also be introduced. For example, perhaps a defense lawyer could find a forensic science article showing that the technique used to link the bullet to the gun by police investigators is unreliable. Or, perhaps there is alibi evidence putting your client on the other side of town at the time that the coroner erroneously believes the time of death to have occurred.
The lawyer would almost always advise the client not to talk to anyone about the case in any respect, so as not to create evidence which could be used against him. Usually, in a case like this one, the safer bet would be to not have the defendant testify (which is by far the more common choice in most criminal cases). If the client testifies, a judge or jury could say that they found him guilty because his demeanor demonstrated guilt and it seemed as if he was lying, but if the client doesn't testify, that kind of finding of fact can't support a conviction.
Any evidence which is not admissible would be objected to (e.g. evidence collected in violation of the 4th Amendment).
In the typical exclusionary rule case (usually involving drugs or other contraband), the cops, the lawyers for both sides and the judges all know that your client is guilty, and a defense lawyer is still entitled to try to exclude the evidence and exonerate the client at trial when it was illegally obtained.
Criminal defense lawyers are not barred from arguing false inferences when representing their clients in criminal cases, they are only barred from offering perjured testimony.
Must the lawyer admit it to the judge or is he not allowed to do so,
and still must help the client?
The lawyer is not allowed to do so and would probably be disbarred if he did without informed client permission.
The lawyer must still help the client.
Just because the defendant is factually guilty doesn't mean that the prosecution has sufficient evidence to prove every element of the offense beyond a reasonable doubt.
If the prosecution can easily prove the case, however, the lawyer's best available course of action is best described in this link, although a client will not always agree to this approach.
The defense cases that are ridiculed on TV or in news reports (e.g. the "Twinkie defense") usually involve guilty defendants who aren't willing to plea guilty whose lawyers come up with the most plausible possible defense under the circumstances, even if it sucks and isn't very likely to be believed. That comes with the territory.